The Roquette category of finite p-groups

Serge Bouc

Abstract : Let p be a prime number. This paper introduces the Roquette category R_{p} of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_{p}, every finite p-group P admits a canonical direct summand ∂P, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_{p} can be described in terms of such edges.

The main motivation for considering this category is that the additive functors from R_{p} to abelian groups are exactly the rational p-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary p-groups : this applies to the representation functors R_{K}, where K is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.

AMS Subject classification : 18B99, 19A22, 20C99, 20J15.

Keywords : p-group, Roquette, rational, biset, genetic.

1. Introduction

Let p be a prime number. This article introduces the Roquette category R_{p} of finite p-groups, which is an additive tensor category with the following properties :

• Every finite p-group can be viewed as an object of R_{p}. The tensor product of two finite p-groups P and Q in R_{p} is the direct product P × Q.

• In R_{p}, any finite p-group has a direct summand ∂P, called the edge

of P, such that

P ≅ ⊕_{N ⊴ P} ∂(P/N) .

Moreover, if the center of P is not cyclic, then ∂P = 0.

• In R_{p}, every finite p-group P decomposes as a direct sum

P ≅ ⊕_{R∈S} ∂R ,

where S is a finite sequence of Roquette groups, i.e. of p-groups of normal p-rank 1, and such a decomposition is essentially unique. Given the group P, such a decomposition can be obtained explicitly from the knowledge of a genetic basis of P (Theorem 3.11 and Proposition 5.14).

• The tensor product ∂P × ∂Q of the edges of two Roquette p-groups P and Q is isomorphic to a direct sum of a certain number ν_{P,Q} of copies of the edge ∂(P ⋄ Q) of another Roquette group (where both ν_{P,Q} and P ⋄ Q are known explicitly - see Theorem 4.20 and Corollary 4.24).

• The additive functors from R_{p} to the category of abelian groups are exactly the rational p-biset functors introduced in [4].

The latter is the main motivation for considering this category : any structural result on R_{p} will provide for free some information on such rational functors for p-groups, e.g. the representation functors R_{K}, where K is a field of characteristic 0 (see [2], [3], and L. Barker’s article [1]), the functor of units of Burnside rings ([6]), or the torsion part of the Dade group ([5]).

In particular, the above results on R_{p} yield isomorphisms describing the structure of some p-groups as objects of this category, and this is enough to compute the evaluations of rational p-biset functors. For example

(D_{8})^{n} ≅ 1 ⊕ (5^{n} − 1) · ∂C_{2}

in R_{2} (Equation 5.25). More generally, Proposition 5.41 gives a formula for (D_{2^{m}})^{n}. A straightforward consequence, applying the functor R_{ℚ}, is the following

Example : For any n ∈ N, the group (D_{8})^{n} has 5^{n} conjugacy classes of cyclic subgroups.

Another important by-product of the above result giving the tensor structure of R_{p} is the explicit description of a genetic basis of a direct product P × Q, in terms of a genetic basis of P and a genetic basis of Q (Theorem 5.20). This allows in particular for a quick computation of the torsion part of the Dade group of some p-groups, e.g. (Theorem 5.36, Assertion 1 and Assertion 3) :

Theorem :

• Let P be an arbitrary finite direct product of groups of order 2 and dihedral 2-groups. Then the Dade group of any factor group of P is torsion free.

• Let n be a positive integer. For any integer m ≥ 4, let P = SD_{2^{m}} be a semidihedral group of order 2^{m}, and let P^{∗n} denote the central product of n-copies of P. Then the torsion part of the Dade group of P^{∗n} is isomorphic to (ℤ/2ℤ)^{2^{(n−1)(m−3)}}.

This also yields similar results on groups of units of Burnside rings of these groups (Remark 5.39), or on representations of central products of p-groups, as in Examples 5.34 and 5.35 :

Example : Let p be a prime, let X be an extraspecial p-group, and let Q be a non-trivial p-group. Let K be a field of characteristic 0. Then Q has the same number (possibly 0) of isomorphism classes of faithful irreducible representations over K as any central product X ∗ Q.

Another possibly interesting phenomenon is that some non-isomorphic p-groups may become isomorphic in the category R_{p}. This means that some non-isomorphic p-groups cannot be distinguished using only rational p-biset functors. When p = 2, there are even examples where this occurs for groups of different orders (Example 5.16). When p > 2, saying that the p-groups P and Q are isomorphic in R_{p} is equivalent to saying that the group algebras ℚP and ℚQ have isomorphic centers (Proposition 5.17).

The category R_{p} is built as follows : consider first the category R_{p}^{♯}, which is the quotient category of the biset category of finite p-groups (in which objects are finite p-groups and morphisms are virtual bisets) obtained by killing a specific element δ in the Burnside group of the Sylow p-subgroup of P GL(3, 𝔽_{p}). Then take idempotent completion, and additive completion of the resulting category.

In particular, this construction relies on bisets, and related functors. Consequently, the paper is organized as follows : Section 2 is a (not so) quick summary of the background on biset functors, Roquette groups, genetic bases of p-groups, and rational p-biset functors. The category R_{p} is introduced in Section 3, and in Section 4, its tensor structure is described. Finally Section 5 gives some examples and applications.

Acknowledgments : Even though the idea of considering the category R_{p} was implicit in [7], the present paper wouldn’t exist without the illuminating conversations I had with Paul Balmer in november 2010, during which I gradually understood I was a kind of Monsieur Jourdain of tensor categories… I wish to thank him for this revelation.

2. Rational p-biset functors

2.1. Biset functors. The biset category C of finite groups is defined as follows :

• The objects of C are the finite groups.

• Let G and H be finite groups. Then

Hom_{C}(G, H) = B(H, G) ,

where B(H, G) denotes the Grothendieck group of the category of finite (H, G)-bisets, i.e. the Burnside group of the group H × G^{op}.

• Let G, H, and K be finite groups. The composition of morphisms

B(K, H) × B(H, G) → B(K, G)

in the category C is the linear extension of the product induced by the product of bisets (V, U) ↦ V ×_{H} U, where V is a (K, H)-biset, and U is an (H, G)-biset.

• The identity morphism of the finite group G is the image in B(G, G) of the set G, endowed with its (G, G)-biset structure given by left and right multiplication.

2.2. Definition : A biset functor is an additive functor from C to the category of abelian groups. A morphism of biset functors is a natural transformation of functors.

Morphisms of biset functors can be composed, and the resulting category of biset functors is denoted by F. It is an abelian category.

2.3. Examples :

1. The correspondence B sending a finite group G to its Burnside group B(G) is a biset functor, called the Burnside functor : indeed B(G) = Hom_{C}(1, G), so B is in fact the Yoneda functor Hom_{C}(1, −).

2. The formalism of bisets gives a single framework for the usual operations of induction, restriction, inflation, deflation, and transport by isomorphism via the following correspondences :

• If H is a subgroup of G, then let Ind_{H}^{G} ∈ B(G, H) denote the set G, with left action of G and right action of H by multiplication.

• If H is a subgroup of G, then let Res_{H}^{G} ∈ B(H, G) denote the set G, with left action of H and right action of G by multiplication.

• If N ⊴ G, and H = G/N, then let Inf_{H}^{G} ∈ B(G, H) denote the set H, with left action of G by projection and multiplication, and right action of H by multiplication.

• If N ⊴ G, and H = G/N, then let Def_{H}^{G} ∈ B(H, G) denote the set H, with left action of H by multiplication, and right action of G by projection and multiplication.

• If φ : G → H is a group isomorphism, then let Iso_{G}^{H} = Iso_{G}^{H}(φ) ∈ B(H, G) denote the set H, with left action of H by multiplication, and right action of G by taking image by φ, and then multiplying in H.

• When H is a subgroup of G, let Defres_{N_{G}(H)/H}^{G} ∈ B(N_{G}(H)/H, G) denote the set H\G, viewed as a (N_{G}(H)/H, G)-biset. It is equal to the composition Def_{N_{G}(H)/H}^{G} ∘ Res_{N_{G}(H)}^{G}.

• When H is a subgroup of G, let Indinf_{N_{G}(H)/H}^{G} ∈ B(G, N_{G}(H)/H) denote the set G/H, viewed as a (G, N_{G}(H)/H)-biset. It is equal to the composition Ind_{N_{G}(H)}^{G} ∘ Inf_{N_{G}(H)/H}^{N_{G}(H)}.

2.4. p-biset functors. From now on, the symbol p will denote a prime number.

2.5. Definition and Notation :

• The biset category C_{p} of finite p-groups is the full subcategory of C consisting of finite p-groups.

• A p-biset functor is an additive functor from C_{p} to the category of abelian groups. A morphism of p-biset functors is a natural transformation of functors.

• p-biset functors form an abelian category F_{p}.

2.6. Roquette p-groups.

2.7. Definition : A finite group G is call a Roquette group if it has normal rank 1, i.e. if all the normal abelian subgroups of G are cyclic.

The Roquette p-groups have been first classified by… Roquette ([14], see also [12]) : these are the cyclic groups, if p > 2. The Roquette 2-groups are the cyclic groups, the generalized quaternion groups, the dihedral and semidihedral groups of order at least 16.

More generally, the p-hyperelementary Roquette groups have been classified by Hambleton, Taylor and Williams (Theorem 3.A.6 of [13]).

The following schematic diagram represents the lattice of subgroups of the dihedral group D_{16}, the quaternion group Q_{16}, and the semi-dihedral group SD_{16} (an horizontal dotted link between two vertices means that the corresponding subgroups are conjugate) :

[figure: •D_{16}; •D_{8}; •C_{8}; •D_{8}; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; D_{16}; •Q_{16}; •Q_{8}; •C_{8}; •Q_{8}; •; •; •; •; •; •; •; Q_{16}]

[figure: •SD_{16}; •D_{8}; •C_{8}; •Q_{8}; •; •; •; •; •; •; •; •; •; •; •; SD_{16}]

These diagrams give a good idea of the general case.

2.8. Definition : Let G be a finite group, of exponent e. An axis of G is a cyclic subgroup of order e in G. An axial subgroup of G is a subgroup of an axis of G.

With these definitions, let us recall without proof the following properties of Roquette p-groups :

2.9. Lemma : Let P be a non-trivial Roquette p-group, of exponent e_{P}.

1. The center of P is cyclic, hence P admits a unique central subgroup

Z_{P} of order p.

2. There exists a non-trivial subgroup Q of P such that Q ∩ Z(P) = 1 if and only if p = 2, and P is dihedral or semidihedral. In this case moreover |Q| = 2, and N_{P} (Q) = QZ_{P}.

3. If P is not cyclic, then p = 2 and e_{P} = |P |/2.

4. There is a unique axis in P, except in the case P ≅ Q_{8}, where there are three of them. Any axis of P is normal in P.

5. If R is a non-trivial axial subgroup of P, then R ≥ Z_{P}, and R ⊴ P. If moreover |R| ≥ p^{2}, then C_{P} (R) is the only axis of P containing R.

Let us also recall the following :

2.10. Lemma : Let P be a finite Roquette p-group. Then there is a unique simple faithful ℚP -module Φ_{P}, up to isomorphism.

Proof : See [8] Proposition 9.3.5.

2.11. Example : Let P be a cyclic group of order p^{m}, and suppose first that m > 1. The algebra ℚP is isomorphic to the algebra A = ℚ[X]/(X^{p^{m}} − 1). As

X^{p^{m}} − 1 = (X^{p^{m−1}} − 1)Ψ_{p^{m}}(X) ,

where Ψ_{p^{m}} denotes the p^{m}-th cyclotomic polynomial, it follows that there is a split exact sequence of A-modules

0 ↦ ℚ[X]/(Ψ_{p^{m}}) → A → ℚ[X]/(X^{p^{m−1}} − 1) → 0 ,

which can be viewed as the following sequence of ℚP -modules

0 → Φ_{P} → ℚP → ℚ(P/Z) → 0 ,

where Z is the unique subgroup of order p of P. It follows that there is an isomorphism of ℚ-algebras

End_{ℚP}Φ_{P} ≅ ℚ[X]/(Ψ_{p^{m}}) ≅ ℚ(ζ_{p^{m}}) ,

where ζ_{p^{m}} is a primitive p^{m}-th root of unity in ℂ.

Now if m = 0, then P = 1, Φ_{P} = ℚ, and End_{ℚP}Φ_{P} ≅ ℚ, also.

2.12. Expansive and genetic subgroups.

2.13. Definition : A subgroup H of a group G is called expansive if for any g ∈ G such that H^{g} ≠ H, the group (H^{g} ∩ N_{G}(H))H/H contains a non trivial normal subgroup of N_{G}(H)/H, i.e. if

g ∈ G − N_{G}(H) ⟹ ⋂_{n∈N_{G}(H)} (H^{gn} ∩ N_{G}(H))H > H .

2.14. Example : If H ⊴ G, then H is expansive in G. More generally, if N_{G}(H) ⊴ G, then H is expansive in G : indeed N_{G}(H^{g}) = N_{G}(H), for any g ∈ G. Hence for g ∈ G − N_{G}(H)

⋂_{n∈N_{G}(H)} (H^{gn} ∩ N_{G}(H))H = (H^{g} ∩ N_{G}(H))H = H^{g} · H > H .

2.15. Notation : When H is a subgroup of the group G, denote by Z_{G}(H) the subgroup of N_{G}(H), containing H, defined by

Z_{G}(H)/H = Z(N_{G}(H)/H) .

The following is an easy consequence of well known properties of pgroups :

2.16. Lemma : [[8] Lemma 9.5.2] Let Q be a subgroup of a finite p-group P. Then Q is expansive in P if and only if

∀g ∈ P, Q^{g} ∩ Z_{P} (Q) ≤ Q ⟹ Q^{g} = Q .

2.17. Example : Let P be a p-group, and let

Δ(P) = {(x, x) ∈ (P × P) | x ∈ P }

denote the diagonal subgroup of (P ×P). Then N_{P ×P} (Δ(P))/Δ(P) ≅ Z(P), and Δ(P) is expansive in (P × P) if and only if

∀x ∈ P, [P, x] ∩ Z(P) = {1} ⟹ x ∈ Z(P) ,

where [P, x] is the set of commutators [y, x] = y^{−1}x^{−1}yx = y^{−1}y^{x}, for y ∈ P. In particular, if [P, P] ≤ Z(P), then Δ(P) is expansive in (P × P).

Proof : Indeed N_{P ×P} (Δ(P)) consists of pairs (a, b) ∈ (P × P) such that ab^{−1} ∈ Z(P). This shows that N_{P ×P} (Δ(P))/Δ(P) ≅ Z(P), and that

Z_{P ×P} (Δ(P)) = N_{P ×P} (Δ(P)) = (1 × Z(P))Δ(P) .

Now Δ(P)^{(u,v)} = Δ(P)^{(1,x)}, for any (u, v) ∈ (P × P), where x = u^{−1}v, and

Δ(P)^{(1,x)} ∩ Z_{P ×P} (Δ(P)) = {(t, t^{x}) | t ∈ P, t^{−1}t^{x} ∈ Z(P)} .

Hence Δ(P)^{(1,x)} ∩ Z_{P ×P} (Δ(P)) ≤ Δ(P) if and only if for any t ∈ P, the assumption t^{−1}t^{x} ∈ Z(P) implies t = t^{x}, i.e. [t, x] = 1, in other words if [P, x] ∩ Z(P) = {1}. Hence Δ(P) is expansive in (P × P) if and only if for any x ∈ P, the assumption [P, x]∩Z(P) = {1} implies (1, x) ∈ N_{P ×P} (Δ(P)), i.e. x ∈ Z(P), as claimed. The last assertion follows trivially.

2.18. Definition : Let Q be a subgroup of the finite p-group P. Then Q is called a genetic subgroup of P if Q is expansive in P and if the group N_{P} (Q)/Q is a Roquette group.

2.19. Definition : Define a relation _{P} on the set of subgroups of the finite p-group P by

Q _{P} R ⇔ ∃g ∈ P, Q^{g} ∩ Z_{P} (R) ≤ R and ^{g}R ∩ Z_{P} (Q) ≤ Q .

2.20. Lemma : Let P be a finite p-group. If Q and R are subgroups of P such that N_{P} (Q) = N_{P} (R) ⊴ P, then

Q _{P} R ⇔ Q =_{P} R .

Proof : Indeed, since P is a p-group, saying that Q^{g} ∩ Z_{P} (R) ≤ R is equivalent to saying that the subgroup (Q^{g} ∩ N_{P} (R))R/R of N_{P} (R)/R contains no non-trivial normal subgroup of N_{P} (R)/R, i.e. that

⋂_{n∈N_{P} (R)} (Q^{gn} ∩ N_{P} (R))R = R .

But if N_{P} (Q) ⊴ P, then N_{P} (Q) = N_{P} (Q^{g}) = N_{P} (R), for any g ∈ G. Hence

⋂_{n∈N_{G}(R)} (Q^{gn} ∩ N_{P} (R))R = Q^{g} · R .

This is equal to R if and only if Q^{g} ≤ R. Similarly ^{g}R ∩ Z_{P} (Q) ≤ Q if and only if ^{g}R ≤ Q. Hence Q^{g} = R.

2.21. Definition : Let G be a group.

1. A section of G is a pair (T, S) of subgroups of G such that S ⊴ T. The

quotient T/S is called the corresponding subquotient of G.

2. Two sections (T, S) and (Y, X) of G are said to be linked (notation

{(T, S) (Y, X)) if \atop S(T ∩ Y) = T, X(T ∩ Y) = Y, T ∩ X = S ∩ Y .}

They are said to be linked modulo G (notation (T, S) _{G} (Y, X)) if there exists g ∈ G such that (T, S) (^{g}Y ,^{g}X).

Observe in particular that if (T, S) _{G} (Y, X), then the corresponding subquotients T/S and Y/X are isomorphic.

2.22. Theorem : Let P be a finite p-group.

1. If S is a genetic subgroup of P, then the module

V (S) = Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)}Φ_{N_{P} (S)/S}

is a simple ℚP -module. Moreover, the functor Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)} induces an isomorphism of ℚ-algebras

End_{QP} V (S) ≅ End_{ℚN_{P} (S)/S}Φ_{N_{P} (S)/S} .

2. If V is a simple ℚP -module, then there exists a genetic subgroup S of P such that V ≅ V (S).

3. If S and T are genetic subgroups of P, then

V (S) ≅ V (T) ⇔ S _{P} T ⇔ (N_{P} (S), S) _{P} (N_{P} (T), T ) .

In particular, if S _{P} T, then N_{P} (S)/S ≅ N_{P} (T)/T. Moreover, the relation _{P} is an equivalence relation on the set of genetic subgroups of P, and the corresponding set of equivalence classes is in one to one correspondence with the set of isomorphism classes of simple ℚP - modules.

Proof : See [8], Theorem 9.6.1.

2.23. Definition : Let P be a finite p-group. A genetic basis of P is a set of representatives of equivalence classes of genetic subgroups of P for the relation _{P} .

2.24. Faithful elements. ([8] Sections 6.2 and 6.3) Let G be a finite group. If N is a normal subgroup of G, recall from Examples 2.3 that Inf_{G/N}^{G} denotes the set G/N, viewed as a (G, G/N)-biset for the actions given by (projection to the factor group and) multiplication in G/N. Similarly Def_{G/N}^{G} denotes the same set G/N, considered as a (G/N, G)-biset.

There is an isomorphism of (G/N, G/N)-bisets

(2.25)  Id_{G/N} ≅ Def_{G/N}^{G} ∘ Inf_{G/N}^{G} .

More generally, if M and N are normal subgroups of G, there is a isomorphism of (G/M, G/N)-bisets

(2.26)  Def_{G/M}^{G} ∘ Inf_{G/N}^{G} = Inf_{G/MN}^{G/M} ∘ Def_{G/MN}^{G/N} .

It follows that if j_{N}^{G} is defined by

j_{N}^{G} = Inf_{G/N}^{G} ∘ Def_{G/N}^{G} ,

then j_{M}^{G} ∘ j_{N}^{G} = j_{MN}^{G}. In particular j_{N}^{G} is an idempotent of B(G, G). Moreover, by a standard orthogonalization procedure, the elements f_{N}^{G} defined for N ⊴ G by

f_{N}^{G} = ∑_{N≤M ⊴ G} μ_{⊴ G}(N, M)j_{M}^{G} ,

where μ_{⊴ G}(N, M) is the Möbius function of the poset of normal subgroups of G, are orthogonal idempotents of B(G, G), and their sum is equal to Id_{G/N}. The idempotent f_{1}^{G} is of special importance :

2.27. Lemma : Let G be a finite group, and N be a normal subgroup of G.

1. f_{N}^{G} = Inf_{G/N}^{G} ∘ f_{1}^{G/N} ∘ Def_{G/N}^{G} in B(G, G).

2. If N ≠ 1, then Def_{G/N}^{G} ∘ f_{1}^{G} = 0 in B(G/N, G), and f_{1}^{G} ∘ Inf_{G/N}^{G} = 0

in B(G, G/N).

Proof : Assertion 1 is Remark 6.2.9 of [8], and Assertion 2 is a special case of Proposition 6.2.6.

If F is a biset functor, the set ∂F (G) of faithful elements of F (G) is defined by

∂F (G) = F (f_{1}^{G})F (G) .

It can be shown ([8] Lemma 6.3.2) that

∂F (G) = ⋂_{1<N ⊴ G} Ker F (Def_{G/N}^{G}) .

2.28. Example : Let F = R_{K} be the representation functor over a field K of characteristic 0. Then for a finite group G, the group ∂R_{K}(G) is the direct summand of R_{K}(G) with basis the set of (isomorphism classes of) faithful irreducible KG-modules.

2.29. Lemma : Let G be a group, and S be a subgroup of G, such that S ∩ Z(G) ≠ 1. Then Defres_{N_{G}(S)/S}^{G}f_{1}^{G} = 0 in B(N_{G}(S)/S, G).

Proof : Set N = S ∩ Z(G), \overline{G} = G/N, and \overline{S} = S/N. Then

Defres_{N_{G}(S)/S}^{G} = Defres\overline{G} ; N_{\overline{G}}(\overline{S})/\overline{S}Def_{G/N}^{G} .

Now Def_{G/N}^{G}f_{1}^{G} = 0 if N ≠ 1, by Lemma 2.27.

2.30. Theorem : [[8] Theorem 10.1.1] Let P be a finite p-group, and B be a genetic basis of P. Then, for any p-biset functor F, the map

I_{B} = ⊕_{S∈B} Indinf_{N_{P} (S)/S}^{P} : ⊕_{S∈B} ∂F (N_{P} (S)/S) → F (P)

is split injective. A left inverse is the map

D_{B} = ⊕_{S∈B} f_{1}^{N_{P} (S)/S} ∘ Defres_{N_{P} (S)/S}^{P} : F (P) → ⊕_{S∈B} ∂F (N_{P} (S)/S) .

One can show ([8] Lemma 10.1.2) that if B and B′ are genetic bases of P, the map I_{B} is an isomorphism if and only if the map I_{B′} is an isomorphism. This motivates the following definition :

2.31. Definition : A p-biset functor F is called rational if for any finite p-group P, there exists a genetic basis B of P such that the map I_{B} is an isomorphism.

So F is rational if and only if for any finite p-group P and any genetic basis B of P, the map I_{B} is an isomorphism.

2.32. Examples :

• The functor R_{ℚ} of rational representations, which sends the finite p-group P to the group R_{ℚ}(P), is a rational p-biset functor. This example is of course the reason for calling rational the p-biset functors of Definition 2.31. This choice has proved rather unfortunate, since the p-biset functor R_{ℂ} of complex representations is also a rational functor… More generally, if K is a field of characteristic 0, then the functor R_{K} is a rational p-biset functor.

• The functor of units of the Burnside ring, sending a p-group P to the group of units B^{×}(P) of its Burnside ring, is a rational p-biset functor (see [6]).

• Let k be a field of characteristic p. The correspondence sending a finite p-group P to the torsion part D_{k}^{t}(P) of the Dade group of P over k is not a biset functor in general, because of phenomenons of Galois twists, but still the maps I_{B} and D_{B} can be defined for D_{k}^{t}, and Theorem 2.30 holds (see [5]).

3. The Roquette category

3.1. Notation : Let π be a projective plane over 𝔽_{p}, and let X denote a Sylow p-subgroup of Aut(π) ≅ PGL(3, 𝔽_{p}). Let 𝕃 be the set of lines of π, and let ℙ be the set of points of π, both viewed as elements of B(X). Let δ = 𝕃 − ℙ ∈ B(X). Equivalently

δ = (X/I − X/IZ) − (X/J − X/JZ) ,

where I and J are non-conjugate non-central subgroups of order p of X, and Z is the center of X.

Let B_{δ} denote the p-biset subfunctor of B generated by δ.

3.2. Remark : When p = 2, the group X is dihedral of order 8, and δ is well-defined up to a sign. When p > 2, the group X is an extraspecial p-group of order p^{3} and exponent p, and there are several possible choices for the element δ. However, in any case, the functor B_{δ} does not depend on the choice of δ.

3.3. Definition : The Roquette category R_{p} of finite p-groups is defined as the idempotent additive completion of the category R_{p}^{♯}, quotient of the biset category C_{p}, defined as follows :

1. the objects of R_{p}^{♯} are the finite p-groups.

2. if P and Q are finite p-groups, then

Hom_{R_{p}^{♯}}(P, Q) = (B/B_{δ})(Q, P)

is the quotient of B(Q × P^{op}) by B_{δ}(Q × P^{op}).

3. the composition in R_{p}^{♯} is induced by the composition of bisets.

4. the identity morphism of the finite p-group P in R_{p}^{♯} is the image of Id_{P}

in (B/B_{δ})(P, P).

3.4. Remark : It was shown in [7] that R_{p}^{♯} is indeed a category. It was also shown there that if p > 2, the functor B_{δ} is equal to the kernel K of the linearization morphism B → R_{ℚ}. It follows that in this case, for any two finite p-groups P and Q

Hom_{R_{p}}(P, Q) ≅ R_{ℚ}(Q × P^{op})

is isomorphic to the Grothendieck group of (ℚQ, ℚP)-bimodules, or, equivalently by the Ritter-Segal theorem, to the Grothendieck group of the subcategory of (ℚQ, ℚP)-permutation bimodules. In other words in this case, the category R_{p} is the full subcategory category of the category considered by Barker in [1], consisting of finite p-groups. The construction of the category R_{p} is also very similar to the construction of the category ℚG-Morita by Hambleton, Taylor, and Williams in [13] (Definition 1.A.4).

In the case p = 2, the situation is more complicated : the functor B_{δ} is a proper subfunctor of the kernel K, and there is a short exact sequence

0 → K/B_{δ} → B/B_{δ} → R_{ℚ} → 0

of p-biset functors. Moreover, for each p-group P, the group (K/B_{δ})(P) is a finite elementary abelian 2-group of rank equal to the number of groups S in a genetic basis of P for which N_{P} (S)/S is dihedral.

3.5. Lemma : The direct product (P, Q) ↦ P × Q of p-groups induces a well defined symmetric monoidal structure on R_{p}^{♯}.

Proof : Let P, P′, Q and Q′ be finite p-groups. If U is a finite (P′, P)-biset and V is a finite (Q′, Q)-biset, then U × V is a (P′ × Q′, P × Q)-biset. This induces a bilinear map

π : B(P′, P) × B(Q′, Q) → B(P′ × Q′, P × Q) ,

and this clearly induces a symmetric monoidal structure on the biset category C_{p}. This induces a monoidal structure on the quotient category if

π(B_{δ}(P′, P), B(Q′, Q)) ⊆ B_{δ}(P′ × Q′, P × Q) .

But this is a consequence of the following : let X be as defined in Notation 3.1, let U be a finite (P′, P ×X)-set, let D be an X-set, and V be a finite (Q′, Q)- biset. Clearly, there is an isomorphism of (P′ × Q′, P × Q)-sets

(U ×_{X} D) × V ≅ (U × V) ×_{X} D ,

where the right action of X on U × V is defined in the obvious way

∀(u, v) ∈ U × V, ∀x ∈ X, (u, v)x = (ux, v) .

The lemma follows.

3.6. Recall that the objects of the idempotent additive completion R_{p} are by definition formal finite sums ⊕_{(P,e)∈P}(P, e) of pairs of the form (P, e), where P

is a finite p-group, and e is an idempotent in Hom_{R_{p}^{♯}}(P, P) = (B/B_{δ})(P, P). A morphism φ : ⊕_{(P,e)∈P}(P, e) → ⊕_{(Q,f)∈Q}(Q, f) in R_{p} is a matrix indexed by

P × Q, where the coefficient φ_{(P,e),(Q,f)} belongs to fHom_{R_{p}^{♯}}(P, Q)e. The composition of morphisms is given by matrix multiplication. In particular :

3.7. Definition and Notation : Let P be a finite p-group.

1. The object (P, Id_{P}) of R_{p} is denoted by P. Similarly, when Q is a finite p-group, and f ∈ B(Q, P), the corresponding morphism from (P, Id_{P}) to (Q, Id_{P}) in the category R_{p} is simply denoted by f.

2. The edge ∂P of P is the pair (P, f_{1}^{P}) of R_{p}.

The category of additive functors from R_{p} to abelian groups is equivalent to the category of additive functors from R_{p}^{♯} to abelian groups. It was shown in [7] that the latter is exactly the category of rational p-biset functors. If F^{♯} is such a functor, then F^{♯} extends to a functor F on R_{p} defined as follows :

F ( ⊕_{(P,e)∈P}(P, e)) = ⊕_{(P,e)∈P} F^{♯}(e)(F^{♯}(P)) ,

with the obvious definition of F (φ) for a morphism φ in the category R_{p}. In particular, with the above notation

F (∂P) = ∂F^{♯}(P) .

In the sequel, we will use the same symbol for F and F^{♯}, writing in particular F (∂P) = ∂F (P).

3.8. Proposition : Let P be a finite p-group. Then

P ≅ ⊕_{N ⊴ P} ∂(P/N)

in the category R_{p}.

Proof : Let

a : P → ⊕_{N ⊴ P} ∂(P/N)

be the direct sum of the morphisms induced by the elements f_{1}^{P/N}Def_{P/N}^{P} of B(P/N, P), and let

b : ⊕_{N ⊴ P} ∂(P/N) → P

be defined similarly from the elements Inf_{P/N}^{P}f_{1}^{P/N} of B(P, P/N). By Lemma 2.27

∑_{N ⊴ P} Inf_{P/N}^{P}f_{1}^{P/N}Def_{P/N}^{P} = ∑_{N ⊴ P} f_{N}^{P} = Id_{P}

in B(P, P), thus a ∘ b is equal to the identity morphism of P in R_{P}. Conversely, for normal subgroups N and M of P

f_{1}^{P/N}Def_{P/N}^{P}Inf_{P/M}^{P}f_{1}^{P/M} = f_{1}^{P/N}Inf_{P/NM}^{P/N}Def_{P/NM}^{P/M}f_{1}^{P/M} ,

by Equation 2.25. This is equal to 0 if N ≠ M, by Lemma 2.27. And if N = M, this is equal to f_{1}^{P/N}. It follows that b ∘ a is equal to the identity morphism of ⊕_{N ⊴ P} ∂(P/N), and this completes the proof.

3.9. Corollary : If P is non-trivial, with cyclic center, then

P ≅ ∂P ⊕ (P/Z)

in R_{p}, where Z is the unique central subgroup of order p in P.

Proof : Indeed, if N is a non-trivial normal subgroup of P, then N ≥ Z. Thus

P ≅ ∂P ⊕ ⊕_{N≥Z} ∂(P/N) ≅ ∂P ⊕ (P/Z)

in the category R_{p}.

3.10. Remark : More generally, let P be a finite p-group, and let N be a normal subgroup of P. Since in R_{p}

P/N ≅ ⊕_{N≤M ⊴ P} ∂((P/N)/(M/N)) ≅ ⊕_{N≤M ⊴ P} ∂(P/M) ,

it follows that P/N is isomorphic to a direct summand of P in the category R_{p}.

3.11. Theorem :

1. The Roquette category R_{p} is an additive tensor category.

2. Let P be a finite p-group, and B be a genetic basis of P. Then, in the

category R_{p},

P ≅ ⊕_{S∈B} ∂\overline{N}_{P} (S) ,

where \overline{N}_{P} (S) = N_{P} (S)/S.

3. Let P be a finite p-group, and B be a genetic basis of P. Then, in the

category R_{p},

∂P ≅ ⊕_{{S∈B \atop S∩Z(P)=1}} ∂\overline{N}_{P} (S) .

Proof : Assertion 1 results from standard results : in particular, the tensor product of ⊕_{(P,e)∈P}(P, e) and ⊕_{(Q,f)∈Q}(Q, f) is defined by

( ⊕_{(P,e)∈P}(P, e)) × ( ⊕_{(Q,f)∈Q}(Q, f)) = ⊕_{{(P,e)∈P \atop (Q,f)∈Q}}(P × Q, e × f) .

For Assertion 2, by Proposition 10.7.2 of [8], if F is a rational p-biset functor, the functor F_{P} obtained from F by the Yoneda-Dress construction at P is also a rational p-biset functor. This applies in particular to the functor Y = B/B_{δ}, so the functor Y_{P} is rational. Hence, if Q is any finite p-group and B_{Q} is a genetic basis of Q, there are mutual inverse isomorphisms

\xymatrix{Y_{P} (Q) \ar[r]^{D_{Q}} & ⊕_{S∈B_{Q}} ∂Y_{P} (\overline{N}_{Q}(S)) \ar[l]^{I_{Q}}}

where I_{Q} = ⊕_{S∈B_{Q}} Indinf_{\overline{N}_{P} (S)}^{P} and D_{Q} = ⊕_{S∈B_{Q}} f_{1}^{\overline{N}_{P} (S)} ∘ Defres_{\overline{N}_{P} (S)}^{P}. Thus for

any f ∈ Y_{P} (Q)

f = ( ∑_{S∈B_{Q}} Indinf_{\overline{N}_{P} (S)}^{P}f_{1}^{\overline{N}_{P} (S)}Defres_{\overline{N}_{P} (S)}^{P}) ∘ f .

Applying this to Q = P, B_{Q} = B, and f = Id_{P} gives that I_{P} ∘ D_{P} = Id_{P}. On the other hand, by Proposition 6.4.4 and Theorem 9.6.1 of [8], for S, T ∈ B, the composition

f_{1}^{\overline{N}_{P} (S)}Defres_{\overline{N}_{P} (S)}^{P} ∘ Indinf_{\overline{N}_{P} (S)}^{P}f_{1}^{\overline{N}_{P} (S)}

is equal to f_{1}^{\overline{N}_{P} (S)} in B(\overline{N}_{P} (S), \overline{N}_{P} (S)) if T = S, and to 0 if T ≠ S. It follows that D_{P} ∘ I_{P} is also equal to the identity map of the direct sum ⊕_{S∈B} ∂Y_{P} (\overline{N}_{P} (S)) in the category R_{p}.

For Assertion 3, observe that by Lemma 2.29

f_{1}^{\overline{N}_{P} (S)}Defres_{\overline{N}_{P} (S)}^{P}f_{1}^{P} = 0

if S ∩ Z(P) ≠ 1. Taking opposite bisets, this gives also

f_{1}^{P} Indinf_{\overline{N}_{P} (S)}^{P}f_{1}^{\overline{N}_{P} (S)} = 0 ,

so the isomorphism of Assertion 2 restricts to an isomorphism

∂P ≅ ⊕_{{S∈B \atop S∩Z(P)=1}} ∂\overline{N}_{P} (S) ,

as the diagram

[figure: P; ≅; ⊕_{S∈B} ∂\overline{N}_{P} (S); ∂P; f_{1}^{P}; ⊕_{{S∈B \atop S∩Z(P)=1}} ∂\overline{N}_{P} (S)]

is commutative.

3.12. Corollary : Let P be a finite p-group. If Z(P) is non-cyclic, then

∂P = 0 in R_{p}.

Proof : This follows from Assertion 3 : suppose indeed that there exists a genetic subgroup S of P such that S ∩ Z(P) ≠ 1. Then the group Z(P) maps injectively in the center of the Roquette group \overline{N}_{P} (S), which is cyclic. Hence Z(P) is cyclic.

3.13. Examples :

1. Let P = D_{8} be a dihedral group of order 8. Let A, B, and C be the subgroups of index 2 in P, and let I be a non-central subgroup of order 2 in P. Then the set {P, A, B, C, I} is a genetic basis of P, and there is an isomorphism

P ≅ 1 ⊕ 4 · ∂C_{2}

in the category R_{2}, where 4 · ∂C_{2} denotes the direct sum of 4 copies of ∂C_{2} : indeed, for S ∈ \{A, B, C, I\}, the group \overline{N}_{P} (S) is isomorphic to C_{2}.

2. Let P = Q_{8} be a quaternion group of order 8. Let A, B, and C be the subgroups of index 2 in P. Then the set {P, A, B, C, 1} is a genetic basis of P (such a basis is unique in this case), and there is an isomorphism

P ≅ 1 ⊕ 3 · ∂C_{2} ⊕ ∂Q_{8}

in the category R_{2}.

3. Let P = (C_{p})^{n} be an elementary abelian p-group of rank n. Then P has a unique genetic basis, consisting of P, and all its subgroups of index p. Hence

P ≅ 1 ⊕ \frac{p^{n} − 1}{p − 1} · ∂C_{p}

in the category R_{p}.

4. The tensor structure

4.1. Notation : Let G and H be groups. When L is a subgroup of G × H,

\begin{aligned}set \\ & p_{1}(L) & = & \{g ∈ G | ∃h ∈ H, (g, h) ∈ L\} \\ & p_{2}(L) & = & \{h ∈ H | ∃g ∈ G, (g, h) ∈ L\} \\ & k_{1}(L) & = & \{g ∈ G | (g, 1) ∈ L\} \\ & k_{2}(L) & = & \{h ∈ H | (1, h) ∈ L\} \end{aligned}

Recall ([8] 2.3.18 and 2.3.21) that k_{i}(L) ⊴ p_{i}(L), for i ∈ {1, 2}, and that (k_{1}(L) × k_{2}(L)) ⊴ L. Set q(L) = L/(k_{1}(L) × k_{2}(L)), and recall that there are canonical group isomorphisms

q(L) ≅ p_{1}(L)/k_{1}(L) ≅ p_{2}(L)/k_{2}(L) .

4.2. Definition : Let G and H be groups. A subgroup L of (G × H) will be called diagonal if

L ∩ (G × 1) = L ∩ (1 × H) = 1 ,

i.e. equivalently, if k_{1}(L) = 1 and k_{2}(L) = 1.

The subgroup L will be called centrally diagonal if

L ∩ (Z(G) × 1) = L ∩ (1 × Z(H)) = 1 ,

i.e. equivalently, if k_{1}(L) ∩ Z(G) = 1 and k_{2}(L) ∩ Z(H) = 1.

4.3. Notation : Let G and H be groups. When K is a subgroup of G, and φ : K → H is a group homomorphism, set

\overrightarrow{Δ}_{φ}(K) = \{(x, φ(x)) | x ∈ K\} ≤ G × H ,

and ; \overleftarrow{Δ}_{φ}(K) = \{(φ(x), x) | x ∈ K\} ≤ H × G .

4.4. Remark : The subgroup L of (G × H) is diagonal if and only if there exists a subgroup K ≤ G and an injective group homomorphism φ : K ↪ H such that L = \overrightarrow{Δ}_{φ}(K).

4.5. Lemma : Let P and Q be p-groups, and let L and L′ be genetic subgroups of (P × Q) such that L _{P ×Q} L′. Then L is centrally diagonal in (P × Q) if and only if L′ is centrally diagonal in (P × Q).

Proof : Let (x, y) ∈ (P × Q) such that

L^{′(x,y)} ∩ Z_{P ×Q}(L) ≤ L .

Since Z(P) × Z(Q) ≤ Z_{P ×Q}(L), it follows that

L′ ∩ (Z(P) × 1) = (L′ ∩ (Z(P) × 1))^{(x,y)}
= L^{′(x,y)} ∩ (Z(P) × 1)
= L^{′(x,y)} ∩ Z_{P ×Q}(L) ∩ (Z(P) × 1)
≤ L ∩ (Z(P) × 1) = 1 .

A similar argument shows that L′ ∩ (1 × Z(Q)) = 1.

Recall from Definition 2.8 that an axial subgroup of a finite group G is a subgroup of a cyclic subgroup of maximal order of G :

4.6. Theorem : Let P and Q be non-trivial Roquette p-groups, let e_{P} (resp. e_{Q}) denote the exponent of P (resp. of Q), and let Z_{P} (resp. Z_{Q}) denote the central subgroup of order p in P (resp. in Q).

1. Let L be a centrally diagonal genetic subgroup of (P × Q). Then L = \overrightarrow{Δ}_{φ}(H), where H ≤ P and φ : H ↪ Q is an injective group homomorphism. Moreover, either P ≅ Q ≅ Q_{8}, and H = P, or H is an axial subgroup of P of order min(e_{P} , e_{Q}), such that φ(H) is an axial subgroup of Q.

2. Conversely :

(a) if P ≅ Q ≅ Q_{8}, let L = L_{φ} = \overrightarrow{Δ}_{φ}(P), where φ : P → Q is a group isomorphism. Then L is a centrally diagonal genetic subgroup of

(P × Q), and N_{P ×Q}(L)/L ≅ C_{2}.

(b) in all other cases, let L = L_{φ} = \overrightarrow{Δ}_{φ}(H), where H is an axial subgroup of P of order min(e_{P} , e_{Q}), and φ : H ↪ Q is an injective group homomorphism such that φ(H) is an axial subgroup of Q. Then L is a centrally diagonal genetic subgroup of (P ×Q). Moreover, the isomorphism class of the group N_{P ×Q}(L)/L depends only

on P and Q.

Proof : • Observe first that the group (Z_{P} ×Z_{Q})L/L is a central subgroup of the Roquette group N_{P ×Q}(L)/L, hence it is cyclic. Hence (Z_{P} ×Z_{Q})∩L ≠ 1. Since both (Z_{P} ×1)∩L and (1×Z_{Q})∩L are trivial, is follows that (Z_{P} ×Z_{Q})∩L is equal to

(4.7)  \overrightarrow{Δ}_{ψ}(Z_{P}) = \{(z, ψ(z)) | z ∈ Z_{P} \} ,

where ψ : Z_{P} →^{≅} Z_{Q} is some group isomorphism. In particular p_{1}(L) contains Z_{P}, and p_{2}(L) contains Z_{Q}.

• Let us prove now that L is diagonal, i.e. that there exists a subgroup H of P and an injective group homomorphism φ : H ↪ Q such that

L = \overrightarrow{Δ}_{φ}(H) = \{(h, φ(h)) | h ∈ H\} .

Otherwise, at least one of the groups k_{1}(L) or k_{2}(L) is non trivial. But the assumption L ∩ (Z(P) × 1) = 1 is equivalent to L ∩ (Z_{P} × 1) = 1, i.e. k_{1}(L) ∩ Z_{P} = 1, and similarly, the assumption L ∩ (1 × Z(Q)) = 1 is equivalent to k_{2}(L) ∩ Z_{Q} = 1. But if there exists a non trivial subgroup X of P such that X ∩ Z_{P} = 1, then p = 2, X has order 2, and P is dihedral or semidihedral (Lemma 2.9). So if L is not diagonal, then p = 2, and at least one of P or Q is dihedral or semidihedral.

Up to exchanging P and Q, one can assume that the group C = k_{1}(L) is non-trivial, hence non-central of order 2 in P. Set A = p_{1}(L). Since A ≤ N_{P} (C) = CZ_{P} (Lemma 2.9), it follows that q(L) = A/C has order 1 or 2.

If q(L) = 1, then A = C, and L = C × D, where D = k_{2}(L) = p_{2}(L). In this case

N_{P ×Q}(L)/L = (N_{P} (C)/C) × (N_{Q}(D)/D) ≅ C_{2} × (N_{Q}(D)/D)

cannot be a Roquette group, since N_{Q}(D)/D is non-trivial (as D ∩ Z_{Q} = 1).

And if |q(L)| = 2, then A = CZ_{P}. If (a, b) ∈ N_{P ×Q}(L), then in particular a ∈ N_{P} (A, C) = A. Thus N_{P ×Q}(L) ≤ A × Q. Now the group N_{P} (A) is a proper subgroup of P, since A is elementary abelian of rank 2, and P is a Roquette group. Choose x ∈ P − N_{P} (A), whence A^{x} ∩ A = Z_{P}. If (a, b) ∈ L^{(x,1)} ∩ (A × Q), then a ∈ A^{x} ∩ A = Z_{P}, hence (a, b) = ^{(x,1)}(a, b) ∈ L. Thus L^{(x,1)} ∩ (A × Q) ≤ L, and

L^{(x,1)} ∩ Z_{P ×Q}(L) ≤ L^{(x,1)} ∩ N_{P ×Q}(L)
≤ L^{(x,1)} ∩ (A × Q)
≤ L

but L^{(x,1)} ≠ L, since A^{x} ≠ A. It follows that L is not expansive in (P × Q), hence L is not a genetic subgroup of (P × Q).

• Hence L is diagonal in (P × Q), i.e.

L = \overrightarrow{Δ}_{φ}(H) = \{(h, φ(h)) | h ∈ H\} ,

for some subgroup H ≥ Z_{P} of P and some φ : H ↪ Q, such that φ(H) ≥ Z_{Q}. Then

N_{P ×Q}(L) = {(x, y) ∈ N_{P} (H) × N_{Q}(φ(H)) | ∀h ∈ H, φ(^{x}h) = ^{y}φ(h)} .

The unique central subgroup of order p of the Roquette group N_{P ×Q}(L)/L is equal to Z/L, where

(4.8)  Z = (Z_{P} × 1)L = (1 × Z_{Q})L .

For any (x, y) ∈ (P × Q), saying that L^{(x,y)} ∩ Z_{P ×Q}(L) is contained in L is equivalent to saying that the group I = L^{(x,y)} ∩ Z is contained in L. In particular, for y = 1

I = L^{(x,1)} ∩ (Z_{P} × 1)L
= {(h^{x}, φ(h)) | h ∈ H, ∃z ∈ Z_{P} , ∃h′ ∈ H, (h^{x}, φ(h)) = (zh′, φ(h′))}
= {(h^{x}, φ(h)) | h ∈ H, h^{−1}h^{x} ∈ Z_{P} } ,

since φ(h) = φ(h′) implies h = h′, and since Z_{P} is central in P. Denoting by [h, x] = h^{−1}h^{x} the commutator of h and x, it follows that I ≤ L if and only if

∀h ∈ H, [h, x] ∈ Z_{P} ⟹ h^{x} ∈ H, (h^{x}, φ(h)) = (h^{x}, φ(h^{x})) .

In other words [h, x] ∈ Z_{P} implies h^{x} = h. Thus I ≤ L if and only if

∀h ∈ H, [h, x] ∈ Z_{P} ⟹ [h, x] = 1 .

Equivalently [H, x] ∩ Z_{P} = {1}, where [H, x] denotes the set of commutators [h, x], for h ∈ H.

Since L = \overrightarrow{Δ}_{φ}(H) is expansive in (P × Q), it follows that

[H, x] ∩ Z_{P} = {1} ⟹ L^{(x,1)} = L .

Now (x, 1) normalizes L if and only if x ∈ C_{P} (H), i.e. if [H, x] = {1}. Hence

(4.9)  [H, x] ∩ Z_{P} = {1} ⟹ [H, x] = {1} .

• Let us show now that unless H = P ≅ Q ≅ Q_{8}, the group H is an axial subgroup of P, and the subgroup φ(H) is an axial subgroup of Q.

Let X be a cyclic subgroup of P of order e_{P}, let x be a generator of X, and suppose that H ≰ X. Then in particular P is not cyclic, so p = 2, the group X is a (normal) subgroup of index 2 of P (Lemma 2.9), and X is equal to its centralizer in P. Moreover |H : H ∩ X| = 2 since H · X = P. The set [H, x] is equal to {1, x^{2}} if P is cyclic or generalized quaternion, or to {1, x^{2+2^{n−2}}} if P is semidihedral : indeed, the image of H in the group of automorphisms of X has order 2, as H ∩ X centralizes X, and H does not. Since Z_{P} is generated by x^{2^{n−2}}, it follows that [H, x] ∩ Z_{P} = {1} if n ≥ 4, i.e. if |P | ≥ 16. But [H, x] ≠ {1}, hence L is not expansive in (P × Q), if P ≥ 16.

So if H ≰ X, then p = 2, and P is non-cyclic, of order at most 8. Hence P ≅ Q_{8}. If H ≠ P, then H is cyclic, and H ≰ X. Thus |H| = 4 = e_{P}, and in particular H is an axis of P. Since H embeds into Q, it follows that |H| = min(e_{P} , e_{Q}). The same argument applied to φ(H) shows that φ(H) is an axial subgroup of Q, as claimed.

In this case moreover, the group Q cannot be isomorphic to Q_{8} : indeed otherwise, one can assume that P = Q and L = Δ(H) is the diagonal embedding. Then

N_{P ×P} (L) = {(a, b) | a^{−1}b ∈ H} .

The group N_{P ×P} (L)/L has order 8, generated by the cyclic subgroup

C = {(a, 1)L | a ∈ H}

of index 2, and the involution (b, b)L, where b ∈ P −H. Hence N_{P ×P} (L)/L ≅ D_{8} is not a Roquette group, and L is not a genetic subgroup of (P × P).

If H is non-cyclic, then H = P, and the same argument applied to φ(H) shows that Q ≅ Q_{8}. And indeed \overrightarrow{Δ}_{φ}(P) is a genetic subgroup of P × Q : this follows from Example 2.17, since the map (x, y) ↦ (x, φ^{−1}(y)) is a group isomorphism from (P × Q) to (P × P), sending \overrightarrow{Δ}_{φ}(P) to Δ(P). Moreover [P, P] ≤ Z(P), and Z(P) has order 2. In particular

(4.10)  N_{P ×Q}(L)/L ≅ C_{2}

does not depend on φ, up to isomorphism. This proves part a) of Assertion 2.

• In the remaining cases L = \overrightarrow{Δ}_{φ}(H), where H is a non-trivial axial subgroup of P, and φ : H ↪ Q is such that φ(H) is an axial subgroup of Q. In particular H is cyclic, and non-trivial. As H ≅ φ(H) ≤ Q, it follows that |H| ≤ min(e_{P} , e_{Q}).

Let C_{e_{P}} be an axis of P containing H, and C_{e_{Q}} be an axis of Q containing φ(Q). Then (C_{e_{P}} × C_{e_{Q}})/Δ_{φ}(H) is an abelian normal subgroup of the Roquette group N_{P ×Q}(L)/L, hence it is cyclic. Thus L = Δ_{φ}(H) is not contained in the Frattini subgroup (C_{e_{P} /p} × C_{e_{Q}/p}) of (C_{e_{P}} × C_{e_{Q}}). In particular p_{1}(L) = C_{e_{Q}}, or p_{2}(L) = C_{e_{Q}}. In other words H = C_{e_{P}}, or φ(H) = C_{e_{Q}}, hence |H| = min(e_{P} , e_{Q}). This completes the proof of Assertion 1.

• Now assume that at least one of the groups P or Q is not isomorphic to Q_{8}. Assume also that e_{P} ≤ e_{Q}, and let H be an axis of P : then H is unique if P ≅ Q̸_{8}, and there are three possibilities for H if P ≅ Q_{8} (Lemma 2.9). In any case H ⊴ P. Let K denote an axial subgroup of Q of order e_{P}. Such a group is unique, except if p = 2, e_{P} = 4, and Q ≅ Q_{8} (thus P ≅ C_{4} as P has exponent 4, and is not isomorphic to Q_{8}). In any case K ⊴ Q.

Let φ : H →^{≅} K be any group isomorphism, and set L_{φ} = \overrightarrow{Δ}_{φ}(H) ≤ (P × Q). Then L_{φ} is obviously centrally diagonal, and

N_{P ×Q}(L_{φ}) = {(a, b) ∈ (P × Q) | ∀h ∈ H, φ(^{a}h) = ^{b}φ(h)} .

Since H is cyclic of order e_{P}, the map

π_{H} : r ∈ (ℤ/e_{P} ℤ)^{×} ↦ (x ↦ x^{r}) ∈ Aut(H)

is a canonical group isomorphism. Similarly, the map

π_{K} : r ∈ (ℤ/e_{P} ℤ)^{×} ↦ (x ↦ x^{r}) ∈ Aut(K)

is a canonical group isomorphism.

Let α : P → Aut(H) ⟶^{π_{H}^{−1}} (ℤ/e_{P} ℤ)^{×} denote the group homomorphism obtained from the action of P on its normal subgroup H by conjugation, and

β : Q → Aut(K) ⟶^{π_{K}^{−1}} (ℤ/e_{P} ℤ)^{×} denote the group homomorphism obtained from the action of Q on its normal subgroup K. Then

N_{P ×Q}(L_{φ}) = {(a, b) ∈ (P × Q) | α(a) = β(b)} .

Now the group (ℤ/e_{P} ℤ)^{×} is abelian. The map

Θ : (a, b) ∈ (P × Q) ↦ β(b)^{−1} · α(a) ∈ (ℤ/e_{P} ℤ)^{×}

is a group homomorphism, and N_{P ×Q}(L_{φ}) = Ker Θ. In particular, it is a normal subgroup of (P × Q), which does not depend on φ, once H and K = φ(H) are fixed. In particular L_{φ} is an expansive subgroup of (P × Q), by Example 2.14. Moreover setting I_{P,Q} = Im(α) ∩ Im(β), there is an exact sequence

(4.11)  1 → C_{P} (H) × C_{Q}(K) → N_{P ×Q}(L_{φ}) ⟶^{Ψ} I_{P,Q} → 1 ,

where Ψ(a, b) = α(a) = β(b), for (a, b) ∈ N_{P ×Q}(L_{φ}).

§. Suppose first that e_{P} = p, i.e. that P ≅ C_{p}. In this case H = Z_{P} = P, and K = Z_{Q}, so L_{φ} is central in (P × Q). Moreover

(4.12)  N_{P ×Q}(L_{φ})/L_{φ} = (P × Q)/L_{φ} = (P × Q)/\overrightarrow{Δ}_{φ}(P) ≅ Q

is a Roquette group, independent of φ, up to isomorphism. In particular L_{φ} is a genetic subgroup of (P × Q).

§. Assume from now on that e_{P} ≥ p^{2}. Then C_{P} (H) ≅ C_{e_{P}}, and C_{Q}(K) ≅ C_{e_{Q}}, by Lemma 2.9.

§§. If p > 2, then P and Q are cyclic, hence H = P, and

(4.13)  N_{P ×Q}(L_{φ})/L_{φ} ≅ (P × Q)/\overrightarrow{Δ}_{φ}(P) ≅ Q

as above. It is a Roquette group, independent of φ, up to isomorphism. In particular L_{φ} is genetic in (P × Q).

§§. Assume now that p = 2. The image of α has order |P : C_{P} (H)|, which is equal to 1 if P is cyclic, and to 2 otherwise. Similarly, the image of β has order |Q : C_{Q}(K)|, which is equal to 1 if K is central in Q, i.e. if Q is cyclic (since |K| = e_{P} ≥ 4 by assumption), and to 2 otherwise. Set I_{P,Q} = Im(α) ∩ Im(β). Then I_{P,Q} has order 1 or 2, and there is an exact sequence

(4.14)  1 → C_{e_{P}} × C_{e_{Q}} → N_{P ×Q}(L_{φ}) → I_{P,Q} → 1 .

Note that I_{P,Q} does not depend on φ : H →^{≅} K : more precisely

Im(α) = \{ \begin{matrix}\{1\} & if P is cyclic, \\ \{1, −1\} & if P is dihedral or generalized quaternion, \\ \{1, e_{P} /2 − 1\} & if P is semidihedral. \end{matrix}

So Im(α) only depends on the type of P.

Similarly

Im(β) = \{ \begin{matrix}\{1\} & if Q is cyclic, \\ \{1, −1\} & if Q is dihedral or generalized quaternion, \\ \{1, e_{Q}/2 − 1\} & if Q is semidihedral. \end{matrix}

Moreover if Q is semidihedral and if e_{Q} > e_{P}, then e_{Q}/2 − 1 ≡ −1(mod.e_{P}), hence Im(β) = {1, −1}. In other words, the group I_{P,Q} is trivial in one of the following cases :

• P or Q is cyclic,

• P is dihedral or generalized quaternion, Q is semidihedral, and e_{P} = e_{Q}, (i.e. equivalently |P | = |Q|),

• P is semidihedral, and Q is dihedral or generalized quaternion,

• P and Q are semidihedral, and e_{P} < e_{Q} (i.e. equivalently |P | < |Q|), and the group I_{P,Q} has order 2 in all other cases, i.e. in one of the following cases :

• P and Q are dihedral or generalized quaternion,

• P is dihedral or generalized quaternion, Q is semidihedral, and |Q|>|P |,

• P and Q are semidihedral, and P ≅ Q.

As L_{φ} ≤ C_{e_{P}} × C_{e_{Q}}, the exact sequence 4.14 yields the exact sequence

(4.15)  1 → (C_{e_{P}} × C_{e_{Q}})/L_{φ} → N_{P ×Q}(L_{φ})/L_{φ} → I_{P,Q} → 1 .

\underline{Case 1 :} If I_{P,Q} is trivial, then N_{P ×Q}(L_{φ}) ≅ C_{e_{P}} × C_{e_{Q}}, and

(4.16)  N_{P ×Q}(L_{φ})/L_{φ} ≅ C_{e_{Q}} ,

which is a Roquette group, independent of φ, up to isomorphism. In particular L_{φ} is a genetic subgroup of (P × Q).

\underline{Case 2 :} Suppose now that I_{P,Q} has order 2, i.e. that Im(α) = Im(β) = \{1, ϵ\}, where ϵ is either −1 or e_{Q}/2−1 (in the case where P and Q are semidihedral and isomorphic). One can choose an element u ∈ P, of order 2 if P is dihedral or semidihedral, and of order 4 if P is generalized quaternion, such that α(u) = ϵ. Similarly, one can choose an element v ∈ Q, of order 2 if Q is dihedral or semidihedral, and of order 4 if Q is generalized quaternion, such that β(v) = ϵ. These choices imply that (u, v) ∈ N_{P ×Q}(L_{φ}).

In the exact sequence 4.15

1 → (C_{e_{P}} × C_{e_{Q}})/L_{φ} → N_{P ×Q}(L_{φ})/L_{φ} → {1, ϵ} → 1 ,

the group C = (C_{e_{P}} × C_{e_{Q}})/L_{φ} is cyclic, isomorphic to C_{e_{Q}}. The element π = (u, v)L_{φ} of N_{P ×Q}(L_{φ})/L_{φ} acts on C in the same way that v acts on the subgroup C_{e_{Q}} of Q, namely by inversion if Q is dihedral or generalized quaternion, and by raising elements to the power e_{Q}/2−1 if Q is semidihedral. Finally π^{2} = (u^{2}, v^{2})L_{φ} = L_{φ} if none of P and Q are generalized quaternion. If P is generalized quaternion and Q is not, then π^{2} = (z_{P} , 1)L_{φ} ∈ C − {1}, where z_{P} is a generator of Z_{P}. Similarly, if Q is generalized quaternion and P is not, then π^{2} = (1, z_{Q})L_{φ} ∈ C −{1}, where z_{Q} is a generator of Z_{Q}. In these two cases π^{4} = (1, 1)L_{φ} = L_{φ}, so π has order 4 in N_{P ×Q}(L_{φ})/L_{φ}. And finally, if both P and Q are generalized quaternion, then π^{2} = (z_{P} , z_{Q})L_{φ} = L_{φ}, since φ(Z_{P}) = Z_{Q}.

(4.17) It follows that the group N_{P ×Q}(L_{φ})/L_{φ} has order 2e_{Q} = |Q|, and that it is :

• dihedral if P and Q are both dihedral, or both generalized quaternion,

• generalized quaternion if one of P, Q is generalized quaternion, and the other is dihedral,

• semidihedral if Q is semidihedral. So N_{P ×Q}(L_{φ})/L_{φ} is a Roquette group, independent of φ, up to isomorphism. In particular, it follows that L_{φ} is a genetic subgroup of (P × Q). This completes the proof of Theorem 4.6.

4.18. Notation :

• Let P and Q be Roquette p-groups. If P and Q are non-trivial, set P ⋄ Q = N_{P ×Q}(L)/L, where L is a centrally diagonal genetic subgroup of (P × Q). Set moreover 1 ⋄ P = P ⋄ 1 = P.

• Let P and Q be Roquette p-groups. If P and Q are non-trivial, let ν_{P,Q} denote the number of equivalence classes of centrally diagonal genetic subgroups of (P × Q) for the relation _{P ×Q}. Set moreover ν_{1,P} =

ν_{P,1} = 1.

4.19. Remark : if P = 1, and Q is a Roquette p-group, then P × Q ≅ Q, and the centrally diagonal genetic subgroups of P × Q are the subgroups 1×R, where R is a genetic subgroup of Q such that R∩Z(Q) = 1. The only such subgroup is R = 1, so N_{Q}(R)/R ≅ Q ≅ N_{P ×Q}(1 × R)/(1 × R). Hence the above definition of P ⋄ Q and ν_{P,Q} is consistent, in the case P = 1.

4.20. Theorem : Let P and Q be Roquette p-groups, of exponents e_{P} and e_{Q}, respectively. Suppose e_{P} ≤ e_{Q}, and set q = |Q|. Then

P ⋄ Q ≅ \{ \begin{matrix}Q & if P = 1 or P ≅ C_{p}, \\ C_{2} & if P ≅ Q ≅ Q_{8}, \\ D_{q} & if q ≥ 16 and P and Q are both dihedral, \\ & or both generalized quaternion, \\ Q_{q} & if one of P, Q is dihedral, \\ & and the other one is generalized quaternion, \\ SD_{q} & if Q is semidihedral, and \\ & - either P is dihedral or generalized quaternion, \\ & and |P | < |Q|, \\ & - or P ≅ Q, \\ C_{e_{Q}} & otherwise. \end{matrix}

Proof : The case P = 1 is trivial, the case P ≅ C_{p} follows from 4.12, the case P ≅ Q ≅ Q_{8} follows from 4.10, the three next cases in the list follow from 4.17, and the last case follows from 4.13 and 4.16.

4.21. Theorem : Let P and Q be Roquette p-groups, of exponent e_{P} and e_{Q}, respectively, and let m = min(e_{P} , e_{Q}).

1. If p = 2 and and one of the groups P or Q is isomorphic to Q_{8}, then then L _{P ×Q} L′ for any centrally diagonal genetic subgroups L and L′ of P × Q. In other words ν_{P,Q} = 1.

2. In all other cases, if L and L′ are centrally diagonal genetic subgroups of P × Q, then L _{P ×Q} L′ if and only if L and L′ are conjugate in

P × Q. In particular ν_{P,Q} = ϕ(m)m\frac{|P ⋄ Q|}{|P ||Q|}, where ϕ is the Euler

function.

Proof : Assume e_{P} ≤ e_{Q}, without loss of generality.

1) If P = 1, then ν_{P,Q} = 1, and P ⋄ Q = Q by definition, and e_{P} = 1, so there is nothing to prove.

2) If p = 2 and P ≅ Q ≅ Q_{8}, then by Theorem 4.6, a genetic centrally diagonal subgroup L of P × Q is of the form L_{φ} = \overrightarrow{Δ}_{φ}(P), where φ : P → Q is some group isomorphism. Moreover N_{P ×Q}(L)/L ≅ C_{2}, so P ⋄ Q ≅ C_{2}.

Now let φ, ψ : P → Q be two group isomorphisms. Then L_{φ} _{P ×Q} L_{ψ} if and only if there exists (x, y) ∈ (P × Q) such that

L_{φ}^{(x,y)} ∩ Z_{P ×Q}(L_{ψ}) ≤ L_{ψ}
(x,y)L_{ψ} ∩ Z_{P ×Q}(L_{φ}) ≤ L_{φ} .

These conditions depend only on the double coset N_{P ×Q}(L_{φ})(x, y)N_{P ×Q}(L_{ψ}), which admits a representative of the form (u, 1).

Now the condition L_{φ}^{(u,1)} ∩ Z_{P ×Q}(L_{ψ}) ≤ L_{ψ} is equivalent to

∀h ∈ P, φ(h)^{−1}ψ(h^{u}) ∈ Z_{Q} ⟹ φ(h)^{−1}ψ(h^{u}) = 1 ,

and similarly, the condition ^{(u,1)}L_{ψ} ∩ Z_{P ×Q}(L_{φ}) ≤ L_{φ} is equivalent to

∀h ∈ P, ψ(h)^{−1}φ(^{u}h) ∈ Z_{Q} ⟹ ψ(h)^{−1}φ(^{u}h) = 1 .

Applying ψ^{−1} to the first condition and φ^{−1} to the second one, and setting θ = ψ^{−1}φ, these two conditions become

∀h ∈ P, θ(h)^{−1}h^{u} ∈ Z_{P} ⟹ θ(h) = h^{u}
∀h ∈ P, θ^{−1}(h)^{−1}^{u}h ∈ Z_{P} ⟹ θ^{−1}(h) = ^{u}h .

Since h^{u}h^{−1} ∈ [P, P] = Z_{P}, there are equivalences

θ(h)^{−1}h^{u} ∈ Z_{P} ⟺ θ(h)^{−1}h ∈ Z_{P} ⟺ h^{−1}θ(h) ∈ Z_{P} ,
θ^{−1}(h)^{−1}^{u}h ∈ Z_{P} ⟺ θ^{−1}(h)^{−1}h ∈ Z_{P} ⟺ h^{−1}θ(h) ∈ Z_{P} .

Hence, in order to prove that L_{φ} _{P ×Q} L_{ψ} for any φ, ψ : P →^{≅} Q, it is enough to prove that (4.22)

∀θ ∈ Aut(P), ∃u ∈ P, ∀h ∈ P, θ(h)h^{−1} ∈ Z_{P} ⟹ \{ \begin{matrix}θ(h) = h^{u} \\ θ^{−1}(h) = ^{u}h \end{matrix} .

If h has order 1 or 2, then θ(h) = h = h^{u} for any u ∈ P, hence the conditions θ(h) = h^{u} and θ^{−1}(h) = ^{u}h only have to be checked for |h| = 4. Now the group Aut(P) permutes the three cyclic subgroups of order 4 of P, and this gives an exact sequence

1 → Inn(P) → Aut(P) → S_{3} → 1 ,

where S_{3} is the symmetric group on three symbols. Saying that θ(h)h^{−1} ∈ Z_{P} is equivalent to saying that θ(<h>) = <h>. Hence, either θ stabilizes the three subgroups of order 4 of P, and in this case θ is inner, hence there exists u ∈ P such that θ(h) = h^{u}, for any h ∈ P, hence θ^{−1}(h) = ^{u}h, for any h ∈ P.

Or there exists a unique subgroup C of order 4 of P such that θ(C) = C. Then either θ(h) = h for any h ∈ C, or θ(h) = h^{−1}, for any h ∈ C. In the first case, take u = 1, and in the second case take u ∈ P − <h>, and then θ(h) = h^{u} for any h ∈ C, hence h = θ^{−1}(h)^{u} for h ∈ C, since C = θ(C). Hence 4.22 holds. This completes the proof in this case.

3) If P ≅ Q_{8} and Q ≇ Q_{8}, then a centrally diagonal genetic subgroup of (P × Q) is of the form L = \overrightarrow{Δ}_{φ}(H), where H is one of the 3 subgroups of order 4 of P, and φ is some isomorphism from H to the unique axial subgroup K of order 4 of Q. Moreover Z_{P ×Q}(L) = L(1 × Z_{Q}).

Let H and H′ be subgroups of order 4 of P. Let φ : H → K and φ′ : H′ → K be group isomorphisms, and set L = \overrightarrow{Δ}(φ) and L′ = \overrightarrow{Δ}_{φ′}(H′). Suppose first that H ≠ H′, and let (a, b) ∈ L′ ∩ Z_{P ×Q}(L) = L(1 × Z_{Q}). It means that a ∈ H′ ∩H = Z_{P}, and that there exists z ∈ Z_{Q} such that φ′(a) = φ(a)z. But the restrictions of φ and φ′ to Z_{P} are equal, so φ′(a) = φ(a), hence z = 1. It follows that L′ ∩ Z_{P ×Q}(L) ≤ L, hence L ∩ Z_{P ×Q}(L′) ≤ L′ by symmetry, so L _{P ×Q} L′ in this case.

Now if H = H′, choose a subgroup H′′ of order 4 in P, different from H, and a group isomorphism φ′′ : H′′ → K. Set L′′ = \overrightarrow{Δ}_{φ′′}(H′′). Then L _{P ×Q} L′′ _{P ×Q} L′, by the previous argument, thus L _{P ×Q} L′.

Hence ν_{P,Q} = 1 in this case, as was to be shown.

4) In the case there are several choices for K, i.e. if Q ≅ Q_{8} and K has order 4 = min(e_{P} , e_{Q}), it follows that P ≅ C_{4}, since P ≇ Q_{8}. In this case, we can exchange P and Q, and use the previous argument. Hence ν_{P,Q} = 1 in this case as well.

5) In all other cases, by Theorem 4.6, a centrally diagonal genetic subgroup L of (P × Q) is of the form \overrightarrow{Δ}_{φ}(H), where H ≤ P is the unique axis of P, and φ : H ↪ Q is a group isomorphism to the unique axial subgroup K of order e_{P} of Q. The normalizer of L in P × Q does not depend on φ, by 4.11, so it does not depend on L, since H and K are also unique.

Let L and L′ be two such centrally diagonal genetic subgroups of P × Q. Then N_{P ×Q}(L) = N_{P ×Q}(L′), thus L _{P ×Q} L′ if and only if L and L′ are conjugate in (P ×Q), by Lemma 2.20. Moreover, it follows from the definition of P ⋄ Q that

|N_{P ×Q}(L) = |L||P ⋄ Q| = e_{P} |P ⋄ Q| ,

so the conjucacy class of L in (P × Q) has cardinality \frac{|P ||Q|}{e_{P} |P ⋄ Q|}. Since

there are ϕ(e_{P}) possible choices for the isomorphism φ : H → K, i.e. ϕ(e_{P}) centrally diagonal subgroups of (P × Q), it follows that

ν_{P,Q} = ϕ(e_{P})e_{P} \frac{|P ⋄ Q|}{|P ||Q|} ,

as was to be shown.

4.23. Remark : Suppose that P ≅ Q_{8} and Q ≇ Q_{8}. Then |P ⋄ Q| = |Q|, by Theorem 4.20. Hence

ϕ(e_{P})e_{P} \frac{|P ⋄ Q|}{|P ||Q|} = 2 × 4 × \frac{|Q|}{8|Q|} = 1 = ν_{P,Q} ,

so the formula for ν_{P,Q} holds in this case. The only case where ν_{P,Q} is not

equal to ϕ(m)m\frac{|P ⋄ Q|}{|P ||Q|} (where m = min(e_{P} , e_{Q})) is the case P ≅ Q ≅ Q_{8} : in this case ν_{P,Q} = 1, but

ϕ(m)m\frac{|P ⋄ Q|}{|P ||Q|} = 2 × 4 × \frac{2}{8 × 8} = \frac{1}{4} .

4.24. Corollary : Let P and Q be Roquette p-groups. Then, in the category R_{p}

∂P × ∂Q ≅ ν_{P,Q} · ∂(P ⋄ Q) .

In other words, if P has exponent e_{P}, if Q has order q and exponent e_{Q}, and if e_{P} ≤ e_{Q} :

∂P ×∂Q = \{ \begin{matrix}∂Q & if P = 1 or P ≅ C_{2}, \\ ∂C_{2} & if P ≅ Q ≅ Q_{8}, \\ \frac{ϕ(e_{P})}{2} · ∂D_{q} & if q ≥ 16 and P and Q are both dihedral, \\ & or both generalized quaternion, \\ \frac{ϕ(e_{P})}{2} · ∂Q_{q} & if one of P, Q is generalized quaternion, \\ & and the other one is dihedral, \\ \frac{ϕ(e_{P})}{2} · ∂SD_{q} & if Q is semidihedral, and \\ & - either P is dihedral or generalized \\ & quaternion, and |P | < |Q|, \\ & - or P ≅ Q, \\ \frac{ϕ(e_{P})e_{P} e_{Q}}{|P ||Q|} · ∂C_{e_{Q}} & otherwise. \end{matrix}

Proof : Let B be a genetic basis of the group R = P × Q. In the category R_{p}, the product ∂P × ∂Q is equal to (P × Q, f_{1}^{P} × f_{1}^{Q}), and it is a summand of R. By Theorem 3.11, there are mutual inverse isomorphisms

\xymatrix{R \ar[r]^{D} & ⊕_{S∈B} ∂\overline{N}_{R}(S) \ar[l]^{I}}

where I is the direct sum of the maps Indinf_{\overline{N}_{R}(S)}^{R}, and D is the direct sum

of the maps f_{1}^{\overline{N}_{R}(S)}Defres_{\overline{N}_{R}(S)}^{R}. Corollary 4.24 follows from the fact that

(4.25)  f_{1}^{\overline{N}_{R}(S)}Defres_{\overline{N}_{R}(S)}^{R}(f_{1}^{P} × f_{1}^{Q}) = 0

unless S is centrally diagonal in R = P × Q : indeed Defres_{\overline{N}_{R}(S)}^{R} is given by the (\overline{N}_{R}(S), R)-biset S\R. On the other hand

f_{1}^{P} × f_{1}^{Q} = (P/1 − P/Z_{P}) × (Q/1 − Q/Z_{Q})
= R/(1 × 1) − R/(1 × Z_{Q}) − R/(Z_{P} × 1) + R/(Z_{P} × Z_{Q}) ,

hence Defres_{\overline{N}_{R}(S)}^{R}(f_{1}^{P} × f_{1}^{Q}) is equal to

S\R/(1 × 1) − S\R/(1 × Z_{Q}) − S\R/(Z_{P} × 1) + S\R/(Z_{P} × Z_{Q})

which is

(4.26)  S\R − S(1 × Z_{Q})\R − S(Z_{P} × 1)\R + S(Z_{P} × Z_{Q})\R .

If S is not centrally diagonal in R = P × Q, then either S = S(1 × Z_{Q}) or S = S(Z_{P} × 1). In each case the sum 4.26 vanishes.

And if S is centrally diagonal in R, then

S(1 × Z_{Q}) = S(Z_{P} × 1) = S(Z_{P} × Z_{Q}) ,

since the image of these groups in the Roquette group \overline{N}_{R}(S) is equal to its unique central subgroup Ŝ/S of order p. In this case

Defres_{\overline{N}_{R}(S)}^{R}(f_{1}^{P} × f_{1}^{Q}) = S\R − Ŝ\R .

Since f_{1}^{\overline{N}_{R}(S)} = N_{R}(S)/S − N_{R}(S)/Ŝ, it follows that Defres_{\overline{N}_{R}(S)}^{R}(f_{1}^{P} × f_{1}^{Q}) is

invariant by composition with f_{1}^{\overline{N}_{R}(S)}.

Conversely, if S is not centrally diagonal in (P × Q), then

(f_{1}^{P} × f_{1}^{Q})Indinf_{\overline{N}_{R}(S)}^{R}f_{1}^{\overline{N}_{R}(S)} = 0 ,

as can be seen by taking opposite bisets in equation 4.25. And if S is centrally diagonal, then

(f_{1}^{P} × f_{1}^{Q})Indinf_{\overline{N}_{R}(S)}^{R}f_{1}^{\overline{N}_{R}(S)} = R/S − R/Ŝ = Indinf_{\overline{N}_{R}(S)}^{R}f_{1}^{\overline{N}_{R}(S)} .

Hence the isomorphisms D and I restrict to mutual inverse isomorphisms between ∂P × ∂Q and the direct sum of the edges ∂\overline{N}_{R}(S), where S is a centrally diagonal genetic subgroup of R. But for all such subgroups S, the group \overline{N}_{R}(S) is isomorphic to P ⋄ Q, and there are ν_{P,Q} centrally diagonal subgroups in a genetic basis of R = P × Q. This completes the proof.

5. Examples and applications

5.1. Suppose first that p is odd. Then the Roquette p-groups are just the cyclic groups C_{p^{n}}, for n ≥ 0. The “multiplication rule” of the edges ∂C_{p^{n}} is the following

(5.2)  ∀m, ∀n ∈ ℕ, ∂C_{p^{m}} × ∂C_{p^{n}} = ϕ(p^{min(m,n)})∂C_{p^{max(m,n)}} ,

where ϕ is the Euler function (thus ϕ(p^{k}) = p^{k−1}(p − 1) if k > 0, and ϕ(1) = 1). 5.3. Some surprising phenomenons occur when p = 2 :

5.4. Proposition : In R_{2}, the edge ∂C_{2} is isomorphic to the trivial group 1 (or its edge ∂1).

Proof : Indeed Corollary 3.12 implies that if E ≅ (C_{2})^{2}, then ∂E = 0 in R_{2}. Let X, Y and Z denote the subgroups of order 2 of E. The element

u = Res_{X}^{E} ×_{E} f_{1}^{E} ×_{E} Ind_{Y}^{E}

of B(X, Y) can be viewed as a morphism from Y to X in the category R_{2}, which factors through ∂E. So this morphism is equal to 0. Since

f_{1}^{E} = E/1 − E/X − E/Y − E/Z + 2E/E ,

it follows that

u = Ind_{1}^{X}Res_{1}^{Y} − Inf_{1}^{X}Res_{1}^{Y} − Ind_{1}^{X}Def_{1}^{Y} − Iso(φ) + 2Inf_{1}^{X}Def_{1}^{Y} ,

where φ is the unique group isomorphism from Y to X. Thus

0 = u Iso(φ^{−1}) = Ind_{1}^{X}Res_{1}^{X} − Inf_{1}^{X}Res_{1}^{X} − Ind_{1}^{X}Def_{1}^{X} − Id_{X} + 2Inf_{1}^{X}Def_{1}^{X} .

Hence in the category R_{2}

Id_{X} = (Ind_{1}^{X} − Inf_{1}^{X})(Res_{1}^{X} − Def_{1}^{X}) + Inf_{1}^{X}Def_{1}^{X} .

It follows that

(5.5)  f_{1}^{X} = f_{1}^{X}(Ind_{1}^{X} − Inf_{1}^{X})(Res_{1}^{X} − Def_{1}^{X})f_{1}^{X} .

But on the other hand f_{1}^{X} = Id_{X} − Inf_{1}^{X}Def_{1}^{X}, so

f_{1}^{X}(Ind_{1}^{X} − Inf_{1}^{X}) = f_{1}^{X}Ind_{1}^{X}
= Ind_{1}^{X} − Inf_{1}^{X}Def_{1}^{X}Ind_{1}^{X}
= Ind_{1}^{X} − Inf_{1}^{X} .

It follows that

(Res_{1}^{X} − Def_{1}^{X})f_{1}^{X}(Ind_{1}^{X} − Inf_{1}^{X}) = (Res_{1}^{X} − Def_{1}^{X})(Ind_{1}^{X} − Inf_{1}^{X})
= 2Id_{1} − Id_{1} − Id_{1} + Id_{1}
= Id_{1} .

Thus, setting

a = f_{1}^{X}(Ind_{1}^{X} − Inf_{1}^{X}) ∈ Hom_{R_{2}}(1, ∂X)
b = (Res_{1}^{X} − Def_{1}^{X})f_{1}^{X} ∈ Hom_{R_{2}}(∂X, 1) ,

the composition b ∘ a is equal to Id_{1}, and Equation 5.5 shows that the composition a ∘ b is equal to the identity of ∂X. So a and b are mutual inverse isomorphisms between 1 and ∂X.

5.6. Corollary : Let F be a rational 2-biset functor. Then for any finite

{2-group P \atop F (C_{2} × P) ≅ F (P) ⊕ F (P) .}

F (D_{8} × P) ≅ F (P)^{⊕5} .

Proof : Indeed rational p-biset functors are exactly those p-biset functors which factor through the category R_{p}. And in the category R_{2}, by Theorem 3.11, there is an isomorphism

C_{2} ≅ 1 ⊕ ∂C_{2} ≅ 1 ⊕ 1 .

Thus C_{2} ×P ≅ P ⊕P, and the first assertion follows. The second one follows from Example 3.13, which shows that in R_{2}

D_{8} ≅ 1 ⊕ 4 · ∂C_{2} ≅ 5 · 1 .

Hence D_{8} × P ≅ 5 · P, thus F (D_{8} × P) ≅ F (P)^{⊕5}.

5.7. Proposition : The edge ∂Q_{8} is an involution : more precisely

∂Q_{8} × ∂Q_{8} = ∂C_{2} ≅ 1 .

Proof : Indeed Q_{8} ⋄ Q_{8} = C_{2}, and ν_{Q_{8},Q_{8}} = 1, by Theorem 4.21.

5.8. Remark : The “action” of this involution on the edges of the other Roquette 2-groups (that is, different from 1, C_{2}, and Q_{8}) is as follows : it stabilizes cyclic and semidihedral groups, and exchanges dihedral and generalized quaternion groups. More precisely, it follows from Corollary 4.24 that

∀n ≥ 2, ∂Q_{8} × ∂C_{2^{n}} = ∂C_{2^{n}}
∀n ≥ 4, ∂Q_{8} × ∂D_{2^{n}} = ∂Q_{2^{n}}
∀n ≥ 4, ∂Q_{8} × ∂Q_{2^{n}} = ∂D_{2^{n}}
∀n ≥ 4, ∂Q_{8} × ∂SD_{2^{n}} = ∂SD_{2^{n}}

5.9. By Theorem 3.11, any finite p-group is isomorphic to a direct sum of edges of Roquette p-groups in the category R_{p}. The following result shows that the summands of such an arbitrary direct sum are unique, up to group isomorphism, with the possible exception of the isomorphism 1 = ∂1 ≅ ∂C_{2} of Proposition 5.4 :

5.10. Proposition : Let S and T be finite sequences of Roquette p-groups, such that there exists an isomorphism

(5.11)  ⊕_{S∈S} ∂S ≅ ⊕_{T ∈T} ∂T

in the category R_{p}. If p = 2, replace any occurrence of C_{2} in S and T by the trivial group, which does not change the existence of the isomorphism 5.11, by Proposition 5.4.

Then there exists a bijection φ : S → T such that the groups S and φ(S) are isomorphic, for any S ∈ S.

Proof : By [3], the simple biset functors S_{R,𝔽_{p}}, where R is a Roquette p-group different from C_{p}, are rational biset functors. Moreover, if |R| ≥ p^{2}, then for any finite p-group P, the dimension of S_{R,𝔽_{p}}(P) is equal to the number of groups S in a genetic basis of P such that \overline{N}_{P} (S) ≅ R. On the other hand, the 𝔽_{p}-dimension of S_{1,𝔽_{p}}(P) is equal to the number of groups S in a genetic basis of P such that |\overline{N}_{P} (S)| ≤ p.

The functor S_{R,𝔽_{p}} extends to an additive functor from R_{p} to the category of 𝔽_{p}-vector spaces, and the value of this functor at the edge ∂P is by definition equal to ∂S_{R,𝔽_{p}}(P). If |R| ≥ p^{2}, then the 𝔽_{p}-dimension of ∂S_{R,𝔽_{p}}(P) is equal to the number of groups S in a genetic basis of P such that \overline{N}_{P} (S) ≅ R and S ∩ Z(P) = 1. In particular, if P itself is a Roquette group, then

dim_{𝔽_{p}} S_{R,𝔽_{p}}(∂P) = dim_{𝔽_{p}} ∂S_{R,𝔽_{p}}(P) = \{ \begin{matrix}1 & if P ≅ R \\ 0 & otherwise \end{matrix}

Applying the functor S_{R,𝔽_{p}} to the isomorphism 5.11, this implies that the number of terms in the sequence S which are isomorphic to R is equal to the corresponding number in the sequence T .

Similarly, for any finite p-group P, the 𝔽_{p}-dimension of ∂S_{1,𝔽_{p}}(P) is equal to the number of groups S in a genetic basis of P such that \overline{N}_{P} (S) ≤ p and S ∩ Z(P) = 1. If P itself is a Roquette group, this gives

dim_{𝔽_{p}} S_{1,𝔽_{p}}(∂P) = dim_{𝔽_{p}} ∂S_{1,𝔽_{p}}(P) = \{ \begin{matrix}1 & if P ≅ C_{p} \\ 1 & if P ≅ 1 \\ 0 & otherwise \end{matrix}

Hence the number of terms in the sequence S which are isomorphic to 1 or C_{p} is equal to the corresponding number in the sequence T . If p = 2, there are no S in S ∪ T such that S ≅ C_{2}, by assumption. It follows that for any Roquette p-group R, the number of terms in the sequence S which are isomorphic to R is equal to the corresponding number in the sequence T . The proposition follows in this case.

If p > 2, the above argument shows that

⊕_{{S∈S \atop |S|≥p^{2}}} ∂S ≅ ⊕_{{T ∈T \atop |T |≥p^{2}}} ∂T .

Let M denote this direct sum. The isomorphism 5.11 can be rewritten as

(5.12)  m_{1}1 ⊕ m_{C_{p}}∂C_{p} ⊕ M ≅ n_{1}1 ⊕ n_{C_{p}}∂C_{p} ⊕ M ,

for some integers m_{1}, m_{C_{p}}, n_{1}, n_{ℂ_{p}} such that m_{1} + m_{C_{p}} = n_{1} + n_{C_{p}}.

Now, let ζ be a primitive (i.e. non-trivial, since p is prime) character (ℤ/pℤ)^{×} → ℂ. Such a character exists since p > 2. The functor S_{C_{p},ζ} is a rational p-biset functor, as it is a summand of ℂR_{ℂ} ([8] Corollary 7.3.5). Applying this functor to the isomorphism 5.12 and taking dimensions gives

m_{C_{p}} + dim_{ℂ} S_{C_{p},ζ}(M) = n_{C_{p}} + dim_{ℂ} S_{C_{p},ζ}(M) ,

since S_{C_{p},ζ}(1) = 0 and S_{C_{p},ζ}(C_{p}) = ∂S_{C_{p},ζ}(C_{p}) ≅ ℂ.

It follows that m_{C_{p}} = n_{C_{p}}, hence m_{1} = n_{1}, which completes the proof.

5.13. Corollary : Let X, Y, and Z be objects of R_{p}, isomorphic to direct sums of edges of Roquette p-groups.

1. If X ⊕ Z ≅ Y ⊕ Z in R_{p}, then X ≅ Y.

2. If n is a positive integer, and if n · X ≅ n · Y in R_{p}, then X ≅ Y.

Proof : Decompose X as X ≅ ⊕_{R} n_{R}(X)·∂R, where R runs through the set of

isomorphism classes of Roquette p-groups, and the function R ↦ n_{R}(X) ∈ ℕ has finite support. Choose similar decompositions Y ≅ ⊕_{R} n_{R}(Y) · ∂R and

Z ≅ ⊕_{R} n_{R}(Z) · ∂R.

For Assertion 1, if p > 2, it follows from Proposition 5.10 that

n_{R}(X) + n_{R}(Z) = n_{R}(Y) + n_{R}(Z) ,

for each R. Thus n_{R}(X) = n_{R}(Y) for each R, hence X ≅ Y in R_{p}.

If p = 2, and if R is a Roquette p-group different from 1 and C_{2}, Proposition 5.10 shows that n_{R}(X) + n_{R}(Z) = n_{R}(Y) + n_{R}(Z), hence n_{R}(X) = n_{R}(Y). Proposition 5.10 also implies that

n_{1}(X) + n_{C_{2}}(X) + n_{1}(Z) + n_{C_{2}}(Z) = n_{1}(Y) + n_{C_{2}}(Y) + n_{1}(Z) + n_{C_{2}}(Z) ,

whence n_{1}(X) + n_{C_{2}}(X) = n_{1}(Y) + n_{C_{2}}(Y), and X ≅ Y in R_{2} again, since 1 ≅ C_{2}.

The proof of Assertion 2 is similar : if p > 2, Proposition 5.10 shows that n n_{R}(X) = n n_{R}(Y), for any R, thus n_{R}(X) = n_{R}(Y), and X ≅ Y. And if p = 2, the conclusion n_{R}(X) = n_{R}(Y) is valid for R different from 1 and C_{2}. Moreover n(n_{1}(X)+n_{C_{2}}(X)) = n(n_{1}(Y)+n_{C_{2}}(Y)), hence n_{1}(X)+n_{C_{2}}(X) = n_{1}(Y) + n_{C_{2}}(Y), and X ≅ Y, since 1 ≅ C_{2}.

In the case of the decomposition of a p-group as a direct sum of edges of Roquette groups, the above isomorphism ∂C_{2} ≅ ∂1 doesn’t matter, and the decomposition is unique :

5.14. Proposition : Let P and Q be finite p-groups. The following assertions are equivalent :

1. The groups P and Q are isomorphic in the category R_{p}.

2. There exist genetic bases B_{P} and B_{Q} of P and Q, respectively, and a bijection σ : B_{P} ⟶^{≅} B_{Q} such that

(5.15)  ∀S ∈ B_{P} , N_{Q}(σ(S))/σ(S) ≅ N_{P} (S)/S .

3. For any genetic bases B_{P} and B_{Q} of P and Q, respectively, there exists a bijection σ : B_{P} ⟶^{≅} B_{Q} such that 5.15 holds.

Proof : Assertion 2 implies Assertion 1 by Theorem 3.11. Now suppose that Assertion 1 holds. Then in particular F (P) ≅ F (Q), for any rational p-biset functor F. Let B_{P} and B_{Q} be genetic bases of P and Q, respectively. If R is a Roquette p-group, set

m_{P} (R) = |{S ∈ B | N_{P} (S)/S ≅ R}| ,

and define similarly m_{Q}(R) for the group Q. The integers m_{P} (R) and m_{Q}(R) do not depend on the choices of the genetic bases B_{P} and B_{Q}.

If R is not isomorphic to C_{p}, then the simple functor S_{R,𝔽_{p}} is rational. Moreover, the 𝔽_{p}-dimension of S_{R,𝔽_{p}}(P) is equal to m_{P} (R) if |R| > p, and to 1 + m_{P} (C_{p}), if R = 1. Since P is the only element S of B such that N_{P} (S)/S = 1, it follows m_{P} (1) = 1, and then m_{P} (R) = m_{Q}(R) for any Roquette p-group R. Assertion 2 follows. The equivalence of Assertions 2 and 3 follows from Theorem 2.22.

5.16. Examples :

• Let p > 2, and let X^{+} (resp. X^{−}) denote the extraspecial p-group of order p^{3} and exponent p (resp. p^{2}). Then X^{+} ≅ X^{−} in R_{p}, for if P is one of these groups, each genetic basis of P consists of S = P, for which N_{P} (S)/S = 1, of the p + 1 subgroups S of index p in P, for which N_{P} (S) = P/S ≅ C_{p}, and an additional non-normal genetic subgroup S such that N_{P} (S)/S ≅ C_{p}. In other words

X^{+} ≅ X^{−} ≅ 1 ⊕ (p + 2) · ∂C_{p}

in the category R_{p}.

• Similar examples exist for p = 2 : if P is one of the groups labelled 6 or 7 in the GAP list of groups of order 32 (see [11]), with respective structure ((C_{4} × C_{2}) ⋊ C_{2}) ⋊ C_{2} and (C_{8} ⋊ C_{2}) ⋊ C_{2}, then in any genetic basis of P, there is a unique group S(= P) such that N_{P} (S)/S = 1, there are 6 groups S such that N_{P} (S)/S ≅ C_{2}, and 2 groups S such that N_{P} (S)/S ≅ C_{4}.

• Some 2-groups with different orders may become isomorphic in the category R_{2} : using GAP, one can show that the elementary abelian group of order 16 is isomorphic to each of the groups labelled 134, 138, and 177 in GAP’s list of groups of order 64. These groups have respective structure

((C_{4} ×C_{4})⋊C_{2})⋊C_{2}, (((C_{4} ×C_{2})⋊C_{2})⋊C_{2})⋊C_{2}, and (C_{2} ×D_{16})⋊C_{2} .

I couldn’t find any similar example for p > 2. In this case however, the following result characterizes those p-groups which become isomorphic in the category R_{p} :

5.17. Proposition : Let p be a prime number, and let P and Q be finite

p-groups.

1. If P ≅ Q in the category R_{p}, then the ℚ-algebras ZℚP and ZℚQ are isomorphic.

2. If p > 2, and if ZℚP and ZℚQ are isomorphic ℚ-algebras, then P ≅ Q in the category R_{p}.

Proof : Let G be a genetic basis of P. For S ∈ G, let V (S) denote the corresponding simple ℚP -module, defined by

V (S) = Indinf_{\overline{N}_{P} (S)}^{P}Φ_{\overline{N}_{P} (S)} .

The multiplicity v_{S} of V (S) in the ℚP -module ℚP is equal to

v_{S} = \frac{dim_{ℚ} V (S)}{dim_{ℚ} End_{ℚP}(V (S))} .

As S is a genetic subgroup of P, there is an isomorphism of (skew-)fields

End_{ℚP}(V (S)) ≅ End_{ℚ\overline{N}_{P} (S)}(Φ_{\overline{N}_{P} (S)}) .

It follows that there is an isomorphism of ℚ-algebras

ℚP ≅ ∏_{S∈G} M_{v_{S}}(End_{ℚ\overline{N}_{P} (S)}(Φ_{\overline{N}_{P} (S)})) .

Hence

ZℚP ≅ ∏_{S∈G} Z(End_{ℚ\overline{N}_{P} (S)}(Φ_{\overline{N}_{P} (S)})) .

It shows that the isomorphism type of the ℚ-algebra ZℚP depends only on the genetic basis G : more precisely, it is determined by the isomorphism type of P in R_{p}. This proves Assertion 1.

Now if p > 2, the group \overline{N}_{P} (S) is cyclic, of order p^{m_{S}}, say. By Example 2.11, there is an isomorphism of (skew-)fields

End_{ℚ\overline{N}_{P} (S)}(Φ_{\overline{N}_{P} (S)}) ≅ ℚ(ζ_{p^{m_{S}}}) ,

where ζ_{p^{m_{S}}} is a primitive root of unity of order p^{m_{S}}. Hence

ZℚP ≅ ∏_{S∈G} ℚ(ζ_{p^{m_{S}}}) .

Similarly, if H is a genetic basis of Q

ZℚQ ≅ ∏_{T ∈H} ℚ(ζ_{p^{n_{T}}}) ,

where p^{n_{T}} = |\overline{N}_{Q}(T)|.

Let l be an integer bigger than all the m_{S}’s, for S ∈ G, and all the n_{T} ’s, for T ∈ H. Set K = ℚ(ζ_{p^{l}}), and let G be the Galois group of K over ℚ. By Galois theory ([15] Theorem 1.5.4 and Remark 1.5.5), the ℚ-algebras ZℚP and ZℚQ are isomorphic if and only if there is an isomorphism of G-sets

Hom_{alg}(ZℚP, K) ≅ Hom_{alg}(ZℚQ, K) .

When r ≤ l is an integer, let G_{r} denote the Galois group of K over ℚ(ζ_{p^{r}}). Then the G-set Hom_{alg}(ZℚP, K) is isomorphic to

⨆_{S∈G} G/G_{n_{S}} .

The isomorphism ZℚP ≅ ZℚQ implies that for any r ≤ l, the number of S ∈ G such that \overline{N}_{P} (S) has order p^{r} is equal to the number of T ∈ H such that \overline{N}_{Q}(T) has order p^{r}. Now Assertion 2 of the proposition follows from Proposition 5.14.

5.18. Remark : Assertion 2 of Proposition 5.17 is not true for p = 2 : let P = D_{8} and Q = Q_{8} denote a dihedral group of order 8 and a quaternion group of order 8, respectively. By Example 3.13, in a genetic basis of P, there is one group S such that N_{P} (S)/S = 1 (namely S = P), and 4 subgroups S such that N_{P} (S)/S ≅ C_{2} (the 3 subgroups of index 2 in P, and a non-central subgroup of order 2 of P). It follows easily that

ℚD_{8} ≅ ℚ ⊕ ℚ ⊕ ℚ ⊕ ℚ ⊕ M_{2}(ℚ) .

On the other hand, a genetic basis of Q contains one subgroup S such that N_{Q}(S)/S = 1 (namely S = Q), 3 subgroups S such that N_{Q}(S)/S ≅ C_{2} (the 3 subgroups of index 2 in Q), and one subgroup S such that N_{Q}(S)/S ≅ Q_{8} (the trivial subgroup of Q). Hence

ℚQ_{8} ≅ ℚ ⊕ ℚ ⊕ ℚ ⊕ ℚ ⊕ ℍ_{ℚ} ,

where ℍ_{ℚ} is the field of quaternions over ℚ. Then

ZℚD_{8} ≅ ℚ^{5} ≅ ZℚQ_{8} .

But D_{8} and Q_{8} are not isomorphic in R_{2}, by Proposition 5.14.

5.19. Genetic bases of direct products. Theorem 4.21 yields a way to compute a genetic basis of a direct products of p-groups. More precisely:

5.20. Theorem : Let P and Q be finite p-groups, let B_{P} be a genetic basis of P, and let B_{Q} be a genetic basis of Q.

1. For each pair (S, T) ∈ B_{P} × B_{Q}, let \overline{R} be a centrally diagonal genetic subgroup of \overline{N}_{P} (S) × \overline{N}_{Q}(T), and let

R = \{(x, y) ∈ N_{P} (S) × N_{Q}(T) | (xS, yT) ∈ \overline{R}\} .

Then R is a genetic subgroup of P × Q, such that

\overline{N}_{P ×Q}(R) ≅ \overline{N}_{P} (S) ⋄ \overline{N}_{Q}(T) .

2. For (S, T) ∈ B_{P} × B_{Q}, let E_{S,T} denote the set of subgroups R obtained in Assertion 1, when \overline{R} runs through a set of representatives of centrally diagonal genetic subgroups of \overline{N}_{P} (S) × \overline{N}_{Q}(T), for the relation \overline{N}_{P} (S)×\overline{N}_{Q}(T) , as described in Theorem 4.21.

Then the sets E_{S,T} consist of mutually inequivalent genetic subgroups of P × Q, for the relation _{P ×Q}, and the (disjoint) union

B_{P ×Q} = ⨆_{(S,T)∈B_{P} ×B_{Q}} E_{S,T}

is a genetic basis of (P × Q).

Proof : Assertion 1 is straightforward if the group \overline{N}_{P} (S) is trivial, i.e. if S = P, or if the group \overline{N}_{Q}(T) is trivial, i.e. if T = Q. So we can assume that S < P and T < Q.

By Theorem 4.6, the group \overline{R} is diagonal in \overline{N}_{P} (S) × \overline{N}_{Q}(T). It follows that k_{1}(R) = S, k_{2}(R) = T, and \overline{R} = R/(S × T). This implies in particular that N_{P ×Q}(R) ≤ N_{P} (S) × N_{Q}(T). More precisely

N_{P ×Q}(R) = \{(a, b) ∈ N_{P} (S) × N_{Q}(T) | (aS, bT) ∈ N_{\overline{N}_{P} (S)×\overline{N}_{Q}(T)}(\overline{R})\} , and the map (a, b) ↦ (aS, bT) induces a group isomorphism

N_{P ×Q}(R)/R ≅ N_{\overline{N}_{P} (S)×\overline{N}_{Q}(T)}(\overline{R})/\overline{R} .

It follows that N_{P ×Q}(R)/R is a Roquette group, and by Theorem 4.6 again, and Notation 4.18

N_{P ×Q}(R)/R ≅ \overline{N}_{P} (S) ⋄ \overline{N}_{Q}(T) .

Let Ŝ ≥ S denote the subgroup of N_{P} (S) such that Ŝ/S is the unique central subgroup of order p of the Roquette group \overline{N}_{P} (S). Define T̂ ≥ T similarly, and let R̂/R be the unique central subgroup of order p of N_{P ×Q}(R)/R. Then

R̂ = (Ŝ × 1)R = (1 × T̂)R .

Let (x, y) ∈ P × Q such that R^{(x,y)} ∩ R̂ ≤ R. Intersecting this inclusion with P × 1 gives

(S^{x} ∩ Ŝ) × 1 ≤ S × 1 ,

thus S^{x} ∩ Ŝ ≤ S, and it follows that x ∈ N_{P} (S), since S is an expansive subgroup of P. Similarly, intersecting the inclusion R^{(x,y)} ∩ R̂ ≤ R with 1 × Q gives T^{y} ∩ T̂ ≤ T, hence y ∈ N_{Q}(T).

Now S × T ≤ R^{(x,y)} ∩ R̂ ≤ R, and taking the quotient by S × T gives

\overline{R}^{(xS,yT)} ∩ (R̂/(S × T)) ≤ \overline{R} .

As \overline{R} is a genetic subgroup of \overline{N}_{P} (S) × \overline{N}_{Q}(T), it follows that \overline{R}^{(xS,yT)} is equal to \overline{R}, hence R^{(x,y)} = R. Thus R is an expansive subgroup of P × Q. Since N_{P ×Q}(R)/R is a Roquette group, the group R is a genetic subgroup of P × Q, and this completes the proof of Assertion 1.

For Assertion 2, let (S, T) and (S′, T′) in B_{P} × B_{Q}, and let R ∈ E_{S,T} and R′ ∈ E_{S′,T′}, such that R _{P ×Q} R′. It means that there exists (x, y) ∈ P × Q such that

(5.21)  R^{(x,y)} ∩ R̂′ ≤ R′ , ^{(x,y)}R′ ∩ R̂ ≤ R .

Intersecting these two inclusions with P × 1 gives

S^{x} ∩ Ŝ′ ≤ S′ , ^{x}S′ ∩ Ŝ ≤ S .

Hence S′ _{P} S, thus S′ = S, since S and S′ are in the same genetic basis of P. Moreover x ∈ N_{P} (S). Similarly, intersecting 5.21 with 1 × Q implies T = T′, and y ∈ N_{Q}(T).

Quotienting the inclusions 5.21 by (S×T) gives that \overline{R}′ _{\overline{N}_{P} (S)×\overline{N}_{Q}(T)} \overline{R}. Hence R′ = R, as was to be shown.

Now setting

B_{P ×Q} = ⨆_{(S,T)∈B_{P} ×B_{Q}} E_{S,T}

yields a set of genetic subgroups of P × Q, which are inequivalent to one other for the relation _{P ×Q}. But

|E_{S,T} | = ν_{\overline{N}_{P} (S),\overline{N}_{Q}(T)} ,

and N_{P ×Q}(R)/R ≅ \overline{N}_{P} (S) ⋄ \overline{N}_{Q}(T), for any R ∈ E_{S,T} it follows that

⊕_{R∈B_{P ×Q}} ∂N_{P ×Q}(R)/R ≅ ⊕_{{S∈B_{P} \atop T ∈B_{Q}}} ν_{\overline{N}_{P} (S),\overline{N}_{Q}(T)}∂(\overline{N}_{P} (S) ⋄ \overline{N}_{Q}(T))
≅ ( ⊕_{S∈B_{P}} ∂\overline{N}_{P} (S)) × ( ⊕_{T ∈B_{Q}} ∂\overline{N}_{Q}(T))
≅ P × Q .

In particular, the rank l_{ℚ}(P × Q) of the group R_{ℚ}(P × Q) is equal to |B_{P ×Q}|. Since B_{P ×Q} is contained in a genetic basis of P × Q, which has cardinality l_{ℚ}(P × Q), it follows that B_{P ×Q} is a genetic basis of P × Q.

5.22. Remark : Theorem 5.20 does not mean that any genetic subgroup of P × Q can be obtained by the construction of Assertion 1. For example, if [P, P] ≤ Z(P) and Z(P) is cyclic, then the diagonal R = Δ(P) is a genetic subgroup of P ×P, by Example 2.17. But k_{1}(R) = 1 is not a genetic subgroup of P, if P is not a Roquette group.

5.23. Example of application. As explained in Example 3.13, the dihedral group D_{8} splits as

(5.24)  D_{8} ≅ 1 ⊕ 4∂C_{2}

in the category R_{2}. By Proposition 5.4, it follows that D_{8} ≅ 5 · 1 in R_{2}. Hence (D_{8})^{n} ≅ 5^{n} · 1, for any n ∈ ℕ. In particular, if F is a rational 2-biset functor such that F (1) = {0}, then F ((D_{8})^{n}) = {0}. Hence F (P) = {0} for any quotient of a direct product of copies of D_{8}, by Remark 3.10.

Actually, one can be more precise : since ∂C_{2} × ∂C_{2} ≅ ∂C_{2} by Corollary 4.24, it follows that for any n ∈ ℕ

(5.25) (D_{8})^{n} ≅ ⊕_{i=0}^{n} \binom{n}{i}4^{i}·(∂C_{2})^{i} = 1⊕ ⊕_{i=1}^{n} \binom{n}{i}4^{i}·∂C_{2} ≅ 1⊕(5^{n}−1)·∂C_{2} .

It means that a genetic basis of the group P = (D_{8})^{n} is made of the group S = P, for which N_{P} (S)/S = 1, and of (5^{n} − 1) subgroups S for which N_{P} (S)/S ≅ C_{2}.

In particular, by Theorem 9.5 of [5] (or Corollary 12.10.3 of [8]), the Dade group of P is torsion free, and so is the Dade group of any factor group of P, by Remark 3.10 again. It shows that the Dade group of a central product of any number of copies of D_{8} is torsion free (see Theorem 5.36 for a generalization of this result) : this was proved by Nadia Mazza and me (Theorem 9.2 of [9]). However, the above argument cannot be considered as a new proof of this result, since Theorem 9.5 of [5] relies on Theorem 9.2 of [9].

5.26. Edges of central products. Let P and Q be non-trivial finite p-groups. Recall that a central product P ∗_{φ} Q of P and Q is by definition a group of the form (P × Q)/\overrightarrow{Δ}_{φ}(Z_{P}), where Z_{P} is a central subgroup of order p of P, and φ : Z_{P} ↪ Z(Q) is some isomorphism from Z_{P} to some central subgroup Z_{Q} of Q.

In the case where p = 2 and the groups P and Q both have cyclic center, the group Z_{P} is unique, as well as the morphism φ, so the central product is simply denoted by P ∗ Q in this case.

5.27. Proposition : Let p be a prime number, and let P and Q be nontrivial finite p-groups. Let Z_{P} (resp. Z_{Q}) denote a central subgroup of order p

of P (resp. Q).

1. If one of the groups Z(P) or Z(Q) is non-cyclic, or if |Z(P)| > p and |Z(Q)| > p, then ∂(P ∗_{φ} Q) = 0 in R_{p}, for any group isomorphism

φ : Z_{P} → Z_{Q}.

2. If Z(P) and Z(Q) are cyclic, and if moreover Z(P) or Z(Q) has order p, then, in R_{p},

⊕_{φ:Z_{P} →^{≅}Z_{Q}} ∂(P ∗_{φ} Q) ≅ ∂P × ∂Q .

Proof : The center of the group P ∗_{φ} Q is equal to Z(P) ∗_{φ} Z(Q). It is cyclic if and only if both Z(P) and Z(Q) are cyclic, and if one of them has order p. This proves Assertion 1.

For Assertion 2, suppose that Z(P) and Z(Q) are cyclic, and that one of them has order p. Then the subgroups Z_{P} and Z_{Q} are uniquely determined, and there are p − 1 group isomorphisms φ : Z_{P} → Z_{Q}. For each of them, the only central subgroup Z_{φ} of order p of P ∗_{φ} Q is equal to (Z_{P} ×Z_{Q})/\overrightarrow{Δ}_{φ}(Z_{P}), and

(P ∗_{φ} Q)/Z_{φ} ≅ (P × Q)/(Z_{P} × Z_{Q}) ≅ \overline{P} × \overline{Q} ,

where \overline{P} = P/Z_{P} and \overline{Q} = Q/Z_{Q}.

By Proposition 3.8

P × Q ≅  ⊕_{1≤N ⊴(P ×Q)} ∂((P × Q)/N)
P ∗_{φ} Q ≅ ⊕_{Δ_{φ}(Z_{P})≤N ⊴(P ×Q)} ∂((P × Q)/N)
\overline{P} × Q ≅ ⊕_{(Z_{P} ×1)≤N ⊴(P ×Q)} ∂((P × Q)/N)
P × \overline{Q} ≅ ⊕_{(1×Z_{Q})≤N ⊴(P ×Q)} ∂((P × Q)/N)
\overline{P} × \overline{Q} ≅ ⊕_{(Z_{P} ×Z_{Q})≤N ⊴(P ×Q)}∂((P × Q)/N) .

Set

(5.28) S = ∂(P × Q) ⊕ ( ⊕_{φ:Z_{P} →^{≅}Z_{Q}}(P ∗_{φ} Q)) ⊕ (\overline{P} × Q) ⊕ (P × \overline{Q}) .

Then S is equal to the direct sum of the edges ∂((P ×Q)/N), for N ⊴(P ×Q), with multiplicity (p + 1) if N ≥ (Z_{P} × Z_{Q}), and multiplicity 1 otherwise. Hence

(5.29)  S ≅ (P × Q) + p · (\overline{P} × \overline{Q}) .

Now ∂(P × Q) = 0 by Corollary 3.12. Moreover, by Corollary 3.9, for each

φ : Z_{P} →^{≅} Z_{Q},

P ∗_{φ} Q = ∂(P ∗_{φ} Q) ⊕ (\overline{P} × \overline{Q}) .

But P ≅ ∂P ⊕ \overline{P} and Q ≅ ∂Q ⊕ \overline{Q}, by Corollary 3.9 again. Replacing P, Q, and P ∗_{φ} Q by these values in Equation 5.28 gives (5.30)

S ≅ ( ⊕_{φ:Z_{P} →^{≅}Z_{Q}} ∂(P ∗_{φ} Q)) ⊕ (p + 1) · (\overline{P} × \overline{Q}) ⊕ (\overline{P} × (∂Q)) ⊕ ((∂P) × \overline{Q}) .

But replacing P by ∂P ⊕ \overline{P} and Q by ∂Q ⊕ \overline{Q} in Equation 5.29 gives

(5.31) S ≅ (∂P × ∂Q) ⊕ (p + 1) · (\overline{P} × \overline{Q}) ⊕ (\overline{P} × (∂Q)) ⊕ ((∂P) × \overline{Q}) .

Comparing Equations 5.30 and 5.31 gives

⊕_{φ:Z_{P} →^{≅}Z_{Q}} ∂(P ∗_{φ} Q) ≅ ∂P × ∂Q ,

by Corollary 5.13. This completes the proof.

5.32. Corollary :

1. Let P and Q be non-trivial finite 2-groups with cyclic center, and assume that Z(P) or Z(Q) has order 2. Then

∂(P ∗ Q) ≅ ∂P × ∂Q

in the category R_{2}.

2. For each i ∈ {1, …, n}, let P_{i} be a finite 2-group with center Z_{i} of

order 2. Let *_{i=1}^{n}P_{i} = P_{1} ∗ P_{2} ∗ ⋯ ∗ P_{n} denote the central product of the

groups P_{i}. Then

*_{i=1}^{n}P_{i} ≅ ∏_{i=1}^{n}(∂P_{i}) ⊕ ∏_{i=1}^{n}(P_{i}/Z_{i})

in the category R_{2}.

3. In particular, for any positive integer n, and any integer m ≥ 4, there are isomorphisms

(D_{2^{m}})^{∗n} ≅ 2^{(n−1)(m−3)} · ∂D_{2^{m}} ⊕ (D_{2^{m−1}})^{n}
(SD_{2^{m}})^{∗n} ≅ 2^{(n−1)(m−3)} · ∂SD_{2^{m}} ⊕ (D_{2^{m−1}})^{n}
(Q_{2^{m}})^{∗n} ≅ \{ \begin{matrix}2^{(n−1)(m−3)} · ∂D_{2^{m}} ⊕ (D_{2^{m−1}})^{n} if n is even \\ 2^{(n−1)(m−3)} · ∂Q_{2^{m}} ⊕ (D_{2^{m−1}})^{n} if n is odd \end{matrix}

in the category R_{2}, where P^{∗n} denote the central product of n copies

of P.

Proof : For Assertion 1, the assumptions imply that there is a unique isomorphism φ : Z_{P} → Z_{Q}. Hence there is only one term in the summation of Proposition 5.27.

Assertion 2 follows from Corollary 3.9, which gives an isomorphism

*_{i=1}^{n}P_{i} ≅ ∂( *_{i=1}^{n}P_{i}) ⊕ (( *_{i=1}^{n}P_{i})/Y ) ,

where Y is the unique central subgroup of order p in *_{i=1}^{n}P_{i}. Now an easy induction argument, using Assertion 1, shows that ∂( *_{i=1}^{n}P_{i}) ≅ ∏_{i=1}^{n}(∂P_{i}), and

that ( *_{i=1}^{n}P_{i})/Y ≅ ∏_{i=1}^{n}(P_{i}/Z_{i}).

Finally, when P is one of the groups D_{2^{m}}, SD_{2^{m}}, or SD_{2^{m}}, then P/Z ≅ D_{2^{m−1}}. Now Assertion 3 follows from Assertion 2 and from an easy induction argument using Corollary 4.24.

5.33. Remark : It follows from Assertion 3 that when n is even, the groups (D_{2^{m}})^{∗n} and (Q_{2^{m}})^{∗n} are isomorphic in the category R_{2} : it is actually easy to check that they are isomorphic as groups.

5.34. Example : From Corollary 4.24 and Assertion 1, it follows that :

• ∂((D_{8})^{∗n}) ≅ ∂C_{2}.

• ∂((Q_{8})^{∗n}) ≅ \{ \begin{matrix}∂C_{2} & if n is even \\ ∂Q_{8} & if n is odd \end{matrix}

• ∂((SD_{2^{m}})^{∗n}) ≅ 2^{(n−1)(m−3)} · ∂SD_{2^{m}}, for m ≥ 4.

More generally, if P is any central product of groups isomorphic to D_{8} or Q_{8}, that is, if P is an extraspecial 2-group, then ∂P ≅ ∂C_{2}, or ∂P ≅ ∂Q_{8}. In particular (see Example 2.28), we recover the well known fact that P has a unique faithful rational irreducible representation. But there is more : let Q be a non trivial 2-group. If the center of Q is not cyclic, then the center of any central product P ∗ Q is not cyclic, hence ∂Q = ∂(P ∗ Q) = 0 in R_{2}. If the center of Q is cyclic, then there is a unique central product P ∗ Q. By Theorem 3.11, there is a finite sequence S of Roquette 2-groups such that

∂Q ≅ ⊕_{R∈S} ∂R

in the category R_{2}. By Assertion 1 of Corollary 5.32, it follows that

∂(P ∗ Q) ≅ ⊕_{R∈S}(∂P × ∂R) .

Now ∂P ≅ ∂C_{2} or ∂P ≅ ∂Q_{8}. In both cases, by Proposition 5.4 and Proposition 5.7, the multiplication by ∂P is a permutation of the edges of the Roquette 2-groups. It follows that there is a sequence S′ of Roquette 2 groups, of the same length as S, such that

∂(P ∗ Q) ≅ ⊕_{R∈S′} ∂R .

In particular, for any field K of characteristic 0, the groups R_{K}(∂P) = ∂R_{K}(P) and R_{K}(∂(P ∗ Q)) = ∂R_{K}(P ∗ Q) are free of the same rank, equal to the length of S or S′.

Hence in any case, the groups Q and P ∗ Q have the same number (possibly 0 if the center of Q is not cyclic) of faithful irreducible representations over K, up to isomorphism.

Similarly, the last example above means in particular that the group (SD_{2^{m}})^{∗n} admits 2^{(n−1)(m−3)} non isomorphic faithful rational irreducible representations.

5.35. Example : Let p be an odd prime, and let P = X^{ϵ} (where ϵ ∈ {±1}) be one of the extraspecial groups of order p^{3} considered in Examples 5.16. Then ∂P ≅ ∂C_{p}. Moreover Z(P) has order p, and any automorphism of Z(P) = Z_{P} can be extended to an automorphism of P. It follows that P ∗_{φ} Q is independent (up to a group isomorphism) of the choice of an embedding φ : P ↪ Z(Q), for any non-trivial p-group Q with cyclic center, so we can denote this group by P ∗ Q.

Now if Q is a non-trivial p-group, and B is a genetic basis of Q, it follows from Theorem 3.11 that

∂Q ≅ ⊕_{{S∈B \atop S∩Z(Q)=1}} ∂\overline{N}_{Q}(S) ,

and the right hand side is a direct sum of non-trivial Roquette p-groups. By Equation 5.2, for any non trivial Roquette p-group R

∂C_{p} × ∂R ≅ (p − 1)∂R .

Hence for any non-trivial p-group Q

∂C_{p} × ∂Q ≅ (p − 1)∂Q .

It follows that if Q is a non-trivial p-group with cyclic center, and if P ≅ X^{ϵ}, then

⊕_{φ:Z_{P} →^{≅}Z_{Q}} ∂(P ∗_{φ} Q) ≅ (p − 1)∂(P ∗ Q) ≅ ∂C_{p} × ∂Q ≅ (p − 1)∂Q ,

hence ∂(X^{ϵ} ∗Q) ≅ ∂Q, by Corollary 5.13. Note that this is also true (for any central product of X^{ϵ} with Q) if the center of Q is non-trivial, since in this case the center of X^{ϵ} ∗ Q is also non trivial, and then ∂(X^{ϵ} ∗ Q) ≅ ∂Q ≅ 0 in R_{p}, by Corollary 3.12.

It follows easily by induction that if P is any central product of groups isomorphic to X^{+} or X^{−}, i.e. if P is an extraspecial p-group, then ∂P ≅ ∂C_{p}. We recover this way the well known fact that for p > 2 also, extraspecial pgroups have a unique faithful rational irreducible representation. The same argument shows more generally that ∂(P ∗ Q) ≅ ∂Q in R_{p}, for any nontrivial p-group Q. In particular Q and P ∗ Q have the same number of faithful irreducible representations over a given field K of characteristic 0.

5.36. Theorem :

1. Let P be an arbitrary finite direct product of groups of order 2 and dihedral 2-groups. Then the Dade group of any factor group of P is torsion free.

2. Let 4 ≤ m_{1} ≤ m_{2} ≤ ⋯ ≤ m_{n} be a non decreasing sequence of integers.

Set s = ∑_{i=1}^{n−1}(m_{i} − 3). Then the torsion part of the Dade group of

*_{i=1}^{n}SD_{2^{m_{i}}} is isomorphic to (ℤ/2ℤ)^{2^{s−1}} if m_{1} < m_{n}, and to (ℤ/2ℤ)^{2^{s}} if

m_{1} = m_{n}.

3. In particular, for any integers n ≥ 1 and m ≥ 4, the torsion part of the Dade group of (SD_{2^{m}})^{∗n} is isomorphic to (ℤ/2ℤ)^{2^{(n−1)(m−3)}}.

Proof : First by Example 3.13

D_{8} ≅ 1 ⊕ 4 · ∂C_{2}

in R_{2}. Now by Corollary 3.9, since the center Z of D_{16} has order 2, and since D_{16}/Z ≅ D_{8}

D_{16} ≅ 1 ⊕ 4 · ∂C_{2} ⊕ ∂D_{16} .

Since for n ≥ 3, the group D_{2^{n}} has a center Z of order 2, and since D_{2^{n}}/Z ≅ D_{2^{n−1}}, it follows by induction that

(5.37)  D_{2^{n}} ≅ 1 ⊕ 4 · ∂C_{2} ⊕ ⨁_{l=4}^{n} ∂D_{2^{l}} .

Now by Corollary 4.24, the product ∂D_{2^{l}} ×∂D_{2^{m}}, for l ≤ m, is isomorphic to 2^{l−3} · ∂D_{2^{m}}. Moreover ∂C_{2} × ∂P ≅ ∂P for any 2-group P, since ∂C_{2} ≅ 1 by Proposition 5.4, it follows that any product of dihedral 2-groups and groups of order 2 is isomorphic to a direct sum of edges of the trivial group, of the edge of the group of order 2, and of edges of dihedral 2-groups.

It follows that if P is a direct product of dihedral 2-groups and groups of order 2, then P is isomorphic in R_{2} to the direct sum of the trivial group, and some copies of edges of the group of order 2 and the edge of dihedral 2-groups. In other words, if S is a genetic subgroup of P, then N_{P} (S)/S is either trivial, or of order 2, or dihedral. Now the Dade group of dihedral 2-groups is torsion free (by [10] Theorem 10.3 (a)), and the Dade groups of the trivial group and the group of order 2 are trivial. It follows that the Dade group of P is torsion free, as well as the Dade group of any quotient of P, as is it a direct summand of the Dade group of P. This proves Assertion 1.

For Assertion 2, set P = *_{i=1}^{n}SD_{2^{m_{i}}}. By Assertion 2 of Corollary 5.32,

(5.38)  P ≅ ∏_{i=1}^{m} ∂SD_{2^{m_{i}}} ⊕ ∏_{i=1}^{n} D_{2^{m_{i}−1}} ,

in the category R_{2}. Now an easy induction on n, using Corollary 4.24, shows that

∏_{i=1}^{n} ∂SD_{2^{m_{i}}} ≅ \{ \begin{matrix}2^{s−1} · ∂C_{2^{mn−1}} & if m_{1} < m_{n} \\ 2^{s} · ∂SD_{2^{mn}} & if m_{1} = m_{n} \end{matrix}

where s = ∑_{i=1}^{n−1}(m_{i} − 3).

By 5.38, this means that in a genetic basis B of P, there are 2^{s−1} or 2^{s} subgroups S such that N_{P} (S)/S is semidihedral, depending on m_{1} < m_{n} or m_{1} = m_{n}, and for the other S ∈ B, the group N_{P} (S)/S is trivial, of order 2, or dihedral.

The Dade group of a dihedral 2-group is torsion free, and the Dade groups of the trivial group and C_{2} are trivial. Moreover, the faithful torsion part ∂D^{t}(C_{2^{m}}) of the Dade group of C_{2^{m}} is isomorphic to ℤ/2ℤ, if m ≥ 2 (see [8] Theorem 12.10.3). Similarly, the faithful torsion part ∂D^{t}(SD_{2^{m}}), for m ≥ 4, is isomorphic to ℤ/2ℤ. This completes the proof of Assertion 2. Assertion 3 is a particular case of Assertion 2.

5.39. Remark : Let P be a finite product of groups of order 2, and dihedral 2-groups, as in Assertion 1, and let Q be a quotient of P. If T is a genetic subgroup of Q, then N_{Q}(T)/T is either trivial, of order 2, or dihedral : indeed T lifts to a genetic subgroup S of P, such that N_{P} (S)/S ≅ N_{Q}(T)/T. It follows in particular that the map

\overline{ϵ}_{Q} : B^{×}(Q) → Hom_{ℤ}(R_{ℚ}(Q), 𝔽_{2})

introduced in [6], Notation 8.4, is a group isomorphism from the group of units of the Burnside ring of Q to the 𝔽_{2}-dual of R_{ℚ}(Q) : indeed, there are non-negative integers a and b_{i}, for i ∈ {4, …, m} such that

Q ≅ 1 ⊕ a · ∂C_{2} ⊕ ⊕_{i=4}^{m} b_{i} · ∂D_{2^{i}}

in the category R_{2}. Then B^{×}(Q) ≅ (𝔽_{2})^{r}, where r = 1 + a + ∑_{i=4}^{m} b_{i}, by [6],

Theorem 8.5. Similarly R_{ℚ}(Q) ≅ ℤ^{r} (hence r is equal to the number of conjugacy classes of cyclic subgroups of Q), so Hom_{ℤ}(R_{ℚ}(Q), 𝔽_{2}) ≅ (𝔽_{2})^{r}. As \overline{ϵ} is injective, it is an isomorphism.

5.40. Proposition : Let m ≥ 3 be an integer. Then for any integer n, there is an isomorphism

(5.41) (D_{2^{m}})^{n} ≅ 1 ⊕ (5^{n} − 1) · ∂C_{2} ⊕ ⨁_{l=4}^{m} \frac{(3 + 2^{l−2})^{n} − (3 + 2^{l−3})^{n}}{2^{l−3}} · ∂D_{2^{l}}

in the category R_{2}.

Proof : Let S_{p} denote the full subcategory of R_{p} consisting of finite direct sums of edges of Roquette p-groups, and let Γ = K_{0}(S_{p}) be the Grothendieck group of this category, for relations given by direct sum decomposition. Then Corollary 5.13 shows that Γ is a free abelian group, and that two objects of S_{p} have the same image in Γ if and only if they are isomorphic in R_{p}. Moreover, by Corollary 4.24, the category S_{p} is a tensor subcategory of R_{p}, and Γ is actually a commutative ring.

It follows that Γ identifies to a subring of the ℚ-algebra ℚΓ = ℚ ⊗_{ℤ} Γ, and that, to prove the proposition, it suffices to check that the two sides of Equation 5.41 have the same image in ℚΓ. Let c denote the image of ∂C_{2} in Γ, and for l ≥ 4, let d_{l} denote the image of ∂D_{2^{l}} in Γ. By Equation 5.37, the image i_{m} of D_{2^{m}} in Γ is equal to

i_{m} = 1 + 4c + ∑_{l=4}^{m} d_{l} .

By Corollary 4.24, for 4 ≤ l ≤ k

d_{l} × d_{k} = 2^{l−3}d_{k} .

It follows that the elements e_{l} = \frac{1}{2^{l−3}} d_{l} of ℚΓ, for l ≥ 4, are such that

∀l, k, 4 ≤ l ≤ k, e_{l} × e_{k} = e_{k} .

In particular e_{l} is an idempotent, and the elements

f_{l} = e_{l} − e_{l+1}, for 4 ≤ l < m, and f_{m} = e_{m}

are orthogonal idempotents of ℚΓ. With this notation, for l ≥ 4

e_{l} = f_{l} + f_{l+1} + ⋯ + f_{m} ,

and the element i_{r} can be written as

i_{m} = 1 + 4c + ∑_{l=4}^{m} d_{l}

= 1 + 4c + ∑_{l=4}^{m} 2^{l−3}(f_{l} + f_{l+1} + ⋯ + f_{m})

= 1 + 4c + ∑_{l=4}^{m} 2(2^{l−3} − 1)f_{l} .

Moreover, it follows from Proposition 5.4 that c × f_{l} = f_{l}, for 4 ≤ l ≤ m. Thus

(i_{m})^{n} = (1 + 4c)^{n} + ∑_{j=1}^{n} \binom{n}{j}(1 + 4c)^{n−j} (∑_{l=4}^{m} 2(2^{l−3} − 1)f_{l})^{j}

= (1 + 4c)^{n} + ∑_{j=1}^{n} \binom{n}{j}5^{n−j} ∑_{l=4}^{m} 2^{j}(2^{l−3} − 1)^{j}f_{l}

= (1 + 4c)^{n} + ∑_{l=4}^{m} (∑_{j=1}^{n} \binom{n}{j}5^{n−j}2^{j}(2^{l−3} − 1)^{j}) f_{l}

(i_{m})^{n} = (1 + 4c)^{n} + ∑_{l=4}^{m} ((5 + 2(2^{l−3} − 1))^{n} − 5^{n})f_{l}

= (1 + 4c)^{n} + ∑_{l=4}^{m} ((3 + 2^{l−2})^{n} − 5^{n})f_{l}

= (1 + 4c)^{n} + ∑_{l=4}^{m} ((3 + 2^{l−2})^{n} − (3 + 2^{l−3})^{n})e_{l} .

The proposition follows, since (1 + 4c)^{n} = 1 + (5^{n} − 1)c, by Equation 5.25,

and since e_{l} = \frac{1}{2^{l−3}} d_{l}.

5.42. Remark : The isomorphism 5.41 is equivalent to saying that a genetic basis of the group P = (D_{2^{m}})^{n} consists of one subgroup S such that N_{P} (S)/S ≅ 1 (namely S = P), of 5^{n} − 1 subgroups S such that

N_{P} (S)/S ≅ C_{2}, and, for 4 ≤ l ≤ m, of \frac{(3 + 2^{l−2})^{n} − (3 + 2^{l−3})^{n}}{2^{l−3}} subgroups

S such that N_{P} (S)/S ≅ D_{2^{l}}.

Together with Assertion 3 of Corollary 5.32, this also gives the structure of genetic bases of the groups (D_{2^{m}})^{∗n}, (SD_{2^{m}})^{∗n}, (Q_{2^{m}})^{∗n} :

5.43. Corollary : Let P be one of the groups D_{2^{m}}, SD_{2^{m}}, or Q_{2^{m}}, for m ≥ 4. Then, for any positive integer n, any genetic basis of the group Q = P^{∗n} consists :

• of one group S such that N_{P} (S)/S = 1 (namely S = Q).

• of 5^{n} − 1 subgroups S such that N_{P} (S)/S ≅ C_{2}.

• for 4 ≤ l ≤ m − 1, of \frac{(3 + 2^{l−2})^{n} − (3 + 2^{l−3})^{n}}{2^{l−3}} subgroups S such that

N_{P} (S)/S ≅ D_{2^{l}}.

• of 2^{(n−1)(m−3)} subgroups S such that N_{P} (S)/S is isomorphic to

D_{2^{m}} if P = D_{2^{m}}
SD_{2^{m}} if P = SD_{2^{m}}
D_{2^{m}} if P = Q_{2^{m}} and n is even
Q_{2^{m}} if P = Q_{2^{m}} and n is odd .

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Serge Bouc - CNRS-LAMFA, Université de Picardie, 33 rue St Leu, 80039, Amiens Cedex 01 - France.

email :  serge.bouc@u-picardie.fr
web   :  http://www.lamfa.u-picardie.fr/bouc/