The Whitehead group of (almost) extra-special p-groups with p odd

Serge Bouc^{1} and Nadia Romero^{2}

Abstract: Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup Cl_{1}(ℤP) of SK_{1}(ℤP), in terms of a genetic basis of P. We also introduce a deflation map Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)), for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK_{1}(ℤP), when P is an elementary abelian pgroup. Keywords: Whitehead group, almost extra-special p-groups, genetic basis. MSC2010: 19B28, 20C05, 20D15.

Introduction

Whitehead groups were introduced by J.H.C. Whitehead in [23], as an algebraic continuation of his work on combinatorial homotopy. The computation of the Whitehead group W h(G) of a finite group G is in general very hard, and a compendium on the subject is the book by Bob Oliver ([17]) of 1988. Since then, it seems that not much progress has been made on this subject (see however [16], [15], [22], [21]).

Let p be an odd prime number. In this paper, we describe the Whitehead group of all extra-special and almost extra-special p-groups. The main reason for focusing on these p-groups is that, apart from elementary abelian p-groups, they are exactly the finite p-groups all proper factor groups of which are elementary abelian (see e.g. Lemma 3.1 of [10]). In particular these groups appear naturally in various areas as first crucial step in inductive procedures (see [10], [9], or the proof of Serre’s Theorem in Section 4.7 of [3] for examples).

One of the main tools in our method is Theorem 9.5 of [17], which gives a first description, for a finite p-group P, of the subgroup Cl_{1}(ℤP), an essential part of the torsion of W h(P). As this description requires the knowledge of the rational irreducible representations of P, it seems natural to try to translate it in terms of a genetic basis of P, which provides an explicit description of the rational irreducible representations of P, using only combinatorial informations on the poset of subgroups of P. The notions of genetic subgroup and genetic basis come from biset functor theory ([7]). They already have been used successfully for the computation of other groups related to representations of finite p-groups, e.g. the group of units of Burnside rings (see [6]), or the Dade group of endopermutation modules (see [5]).

1CNRS-LAMFA, UPJV, Amiens, France

2DEMAT, UGTO, Guanajuato, Mexico - Partially supported by SEP-PRODEP, project PTC-486

Our first task is then to obtain a description of Cl_{1}(ℤP) in terms of a genetic basis of P. The advantage of this description is that it only requires data of a combinatorial nature, and makes no use of linear representations. In particular this makes it much easier to implement for computational purposes, using e.g. GAP software ([13]). This also explains why in the proof of our main theorem (Theorem A below), we have to consider separately two special cases of “small” p-groups, for which the lattice of subgroups is in some sense too tight for the general argument to work.

We observe next on this description that for any normal subgroup N of P, there is an obvious deflation operation Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)). Even if there seems to be no inflation operation in the other direction, which would provide a section of this map, we show that Def_{P/N}^{P} is always surjective. The existence of such a surjective deflation map already follows from Corollary 3.10 of [17], and it is shown in [4] that our deflation map is indeed the same as the map defined there.

As we will see in the next section, in the case of an extra-special or almost extraspecial p-group P, the group Cl_{1}(ℤP) is equal to the torsion part SK_{1}(ℤP) of W h(P), so the computation of the Whitehead group of P comes down to the knowledge of Cl_{1}(ℤ(P/Φ(P))), where Φ(P) is the Frattini subgroup of P, and a detailed analysis of what happens with the unique faithful rational irreducible representation of P. Our main theorem is the following: Theorem A. Let p be an odd prime, and let P be an extra-special p-group of order at least p^{5} or an almost extra-special p-group of order at least p^{6}. Set N = Φ(P) = P′.

1. The group Cl_{1}(ℤP) is isomorphic to K × (C_{p})^{M}, where M = \frac{p^{k−1}−1}{p−1} − \binom{p+k−2}{p} if |P | = p^{k}, and K is the kernel of Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)).

2. The group K is cyclic. More precisely, it is:

(a) trivial if P is extra-special of order p^{5} and exponent p^{2}.

(b) of order p if P is almost extra-special of order p^{6}.

(c) isomorphic to Z(P) in all other cases.

It should be noted that our methods not only give the algebraic structure of the Whitehead group, but allow also for the determination of explicit generators of its torsion subgroup (see. Remark 3.18 for details).

The paper is organized as follows: Section 1 is a review of definitions and basic results on Whitehead groups, genetic bases of p-groups, extra-special and almost extra-special p-groups. In particular, the subgroups Cl_{1}(ℤG) and SK_{1}(ℤG) of the Whitehead group W h(G) of a group G are introduced. In Section 2, we give a procedure (Theorem 2.4) to compute Cl_{1}(ℤP) for a finite p-group P (with p odd) in terms of a genetic basis of P. This procedure may be of independent interest, in particular from an algorithmic point of view. Finally, in Section 3, we begin by giving a new proof of the structure of W h(P) for an elementary abelian p-group P, for p odd, and then we come to our main theorems. The first of them is Theorem 3.16, which will give us point 1 of Theorem A, then Theorem 3.19 and Proposition 3.22 will deal with the description of the group K, giving us point 2. With these results we completely determine the structure of W h(P), when P is an extra-special or almost extra-special p-group, for p odd.

1. Preliminaries

Let G be a group. We will write Z(G) for the center of G and G′ for its commutator subgroup. The Frattini subgroup of G is denoted by Φ(G). If H is a subgroup of G, the normalizer of H in G will be denoted by N_{G}(H), and its centralizer by C_{G}(H). If H = ⟨h⟩ for an element h ∈ G, we may also write C_{G}(H) = C_{G}(h).

1.1. About the Whitehead group.

Let R be an associative ring with unit. The infinite general linear group of R, denoted by GL(R), is defined as the direct limit of the inclusions GL_{n}(R) → GL_{n+1}(R) as the upper left block matrix. If, for each n > 0, we denote by E_{n}(R) the subgroup of GL_{n}(R) generated by all elementary n×n-matrices – i.e. all those which are the identity except for one non-zero off-diagonal entry – and we take the direct limit as before for the groups E_{n}(R), we obtain a subgroup of GL(R), denoted by E(R). Whitehead’s Lemma (Theorem 1.13 in [17], for instance) states that E(R) is equal to the derived subgroup of GL(R). The group K_{1}(R) is defined as the abelianization of GL(R), which is then equal to GL(R)/E(R).

If G is a group and we take R as the group ring ℤG, then elements of the form ±g for g ∈ G can be regarded as invertible 1 × 1-matrices over ℤG and hence they represent elements in K_{1}(ℤG). Let H be the subgroup of K_{1}(ℤG) generated by classes of elements of the form ±g with g ∈ G. The Whitehead group of G is defined as W h(G) = K_{1}(ℤG)/H.

If G is finite, the groups K_{1}(ℤG) and W h(G) are finitely generated abelian groups (see Theorem 2.5 in [17]).

For the rest of this section, G denotes a finite group. 1.2. Definition. Let D be an integral domain and K be its field of fractions, then SK_{1}(DG) denotes the kernel of the morphism

K_{1}(DG) → K_{1}(KG).

By Theorem 7.4 in [17], SK_{1}(ℤG) is isomorphic to the torsion subgroup of W h(G). Hence, W h(G) is completely determined by SK_{1}(ℤG) and the rank of its free part (i.e. its free rank). According to Theorem 2.6 in [17], this free rank is equal to r − q, where r is the number of non-isomorphic irreducible ℝ-representations of G and q is the number of non-isomorphic irreducible ℚ-representations of G. 1.3. Definition. Consider the ring of p-adic integers ℤ̂_{p}. The group Cl_{1}(ℤG) is defined as the kernel of the localization morphism

l : SK_{1}(ℤG) → ⨁_{p} SK_{1}(ℤ̂_{p}G).

By Theorem 3.9 in [17], SK_{1}(ℤ̂_{p}G) is trivial whenever p does not divide |G|, and l is onto. In particular, SK_{1}(ℤG) sits in an extension

0 ⟶ Cl_{1}(ℤG) ⟶ SK_{1}(ℤG) ⟶ ⨁_{p} SK_{1}(ℤ̂_{p}G) ⟶ 0.

This extension is used by Oliver in [17] to describe SK_{1}(ℤG) in many cases. The important feature of the groups treated in this paper is that their derived subgroups are central and so by Theorem 8.10 in [17], the group ⨁_{p} SK_{1}(ℤ̂_{p}G) is trivial, then describing SK_{1}(ℤG) amounts to describing Cl_{1}(ℤG).

1.4. About genetic bases.

A finite p-group Q is called a Roquette p-group if it has normal p-rank 1, i.e. if all its abelian normal subgroups are cyclic. The Roquette p-groups (see [19] or Theorem 4.10 of [14] for details) of order p^{n} are the cyclic groups C_{p^{n}}, if p is odd, and the cyclic groups C_{2^{n}}, the generalized quaternion groups Q_{2^{n}}, for n ⩾ 3, the dihedral groups D_{2^{n}}, for n ⩾ 4, and the semidihedral groups SD_{2^{n}}, for n ⩾ 4, if p = 2. A Roquette p-group Q admits a unique faithful rational irreducible representation Φ_{Q} (see e.g. Proposition 9.3.5 in [7]).

1.5. Definition. Let P be a finite p-group. A subgroup S of P is called genetic if the section S ⊴ N_{P} (S) ⩽ P satisfies

1. The group N_{P} (S)/S is a Roquette group.

2. Let Φ = Φ_{N_{P} (S)/S} be the only faithful irreducible ℚ-representation of N_{P} (S)/S and V = Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)}ϕ, then the functor Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)} induces an isomorphism of ℚ-algebras

End_{ℚP} V ≅ End_{ℚ(N_{P} (S)/S)}Φ.

Note that the right-hand side algebra is actually a skew field by Schur’s lemma. So End_{ℚP} V is also a skew field, hence V is an indecomposable – that is, irreducible – ℚP -module.

1.6. Notation. Let P be a finite p-group and S be a genetic subgroup of P. We write

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

Then V (S) is an irreducible ℚ-representation of P. Conversely, by Roquette’s Theorem (Theorem 9.4.1 in [7]), for each irreducible ℚ-representation V of P, there exists a genetic subgroup S of P such that V ≅ V (S).

The following theorem characterizes the genetic subgroups of a p-group. First some notation: for a subgroup S of a finite p-group P, let Z_{P} (S) ⩾ S be the subgroup of N_{P} (S) defined by Z_{P} (S)/S = Z(N_{P} (S)/S). In particular Z_{P} (S) = N_{P} (S) if N_{P} (S)/S abelian, e.g. if N_{P} (S)/S is a Roquette p-group for p odd.

1.7. Theorem. [Theorem 9.5.6 in [7]] Let P be a finite p-group and S be a subgroup of P such that N_{P} (S)/S is a Roquette group. Then the following conditions are equivalent:

1. The subgroup S is a genetic subgroup of P.

2. If x ∈ P is such that ^{x}S ∩ Z_{P} (S) ⩽ S, then ^{x}S = S.

3. If x ∈ P is such that ^{x}S ∩ Z_{P} (S) ⩽ S and S^{x} ∩ Z_{P} (S) ⩽ S, then ^{x}S = S.

The next result is part of Theorem 9.6.1 in [7].

1.8. Theorem. Let P be a finite p-group and S and T be genetic subgroups of P. The following conditions are equivalent:

1. The ℚP -modules V (S) and V (T) are isomorphic.

2. There exist x, y ∈ P such that ^{x}T ∩ Z_{P} (S) ⩽ S and ^{y}S ∩ Z_{P} (T) ⩽ T.

3. There exists x ∈ P such that ^{x}T ∩ Z_{P} (S) ⩽ S and S^{x} ∩ Z_{P} (T) ⩽ T.

If these conditions hold, then in particular the groups N_{P} (S)/S and N_{P} (T)/T are isomorphic.

The relation between groups appearing in point 2 is denoted by S _{P} T. The theorem shows that this relation is an equivalence relation on the set of genetic subgroups of P, and we have the following definition.

1.9. Definition. [Definition 9.6.11 in [7]] Let P be a finite p-group. A genetic basis of P is a set of representatives of the equivalence classes of _{P} in the set of genetic subgroups of P.

1.10. Lemma. Let P be a finite p-group and S be a genetic subgroup of P.

1. The kernel of V (S) is equal to the intersection of the conjugates of S in P.

2. In particular V (S) is faithful if and only S intersects Z(P) trivially.

Proof. Denote by Φ the unique rational irreducible representation of the Roquette group N_{P} (S)/S. Then

V (S) = Ind_{N_{P} (S)}^{P}Inf_{N_{P} (S)/S}^{N_{P} (S)}Φ ≅ ⨁_{x∈[P/N_{P} (S)]} x ⊗ Φ,

where [P/N_{P} (S)] is a chosen set of representatives of N_{P} (S)-cosets in P. An element g ∈ P acts trivially on V (S) if and only if it permutes trivially the summands of this decomposition, that is if g^{x} ∈ N_{P} (S) for any x ∈ P, and if moreover g^{x} acts trivially on Φ, which means that g^{x} ∈ S, since Φ is faithful. This proves Assertion 1.

Assertion 2 follows, since

( ⋂_{x∈P} ^{x}S) ∩ Z(P) = ⋂_{x∈P} (^{x}S ∩ Z(P)) = ⋂_{x∈P} ^{x}(S ∩ Z(P)) = S ∩ Z(P),

and since the normal subgroup T = ⋂_{x∈P} ^{x}S of P is trivial if and only if T ∩ Z(P) = 1,

i.e. if S ∩ Z(P) = 1.

1.11. Remark. If P is abelian, then a subgroup S of P is genetic if and only if P/S is cyclic. Moreover the relation _{P} is the equality relation in this case, so there is a unique genetic basis of P, consisting of all the subgroups S of P such that P/S is cyclic.

1.12. Remark. Let P be a finite p-group, and S be a subgroup of P such that N_{P} (S) is normal in P. If N_{P} (S)/S is a Roquette group, then S is a genetic subgroup of P. Indeed if N_{P} (S) is normal in P, then N_{P} (S) = N_{P} (^{x}S) for any x ∈ P. Hence xS ⊴ N_{P} (S), and the group ^{x}S · S/S is a normal subgroup of N_{P} (S)/S. It is trivial if and only if it intersects trivially the center Z_{P} (S)/S of N_{P} (S)/S. Then

xS = S ⟺ ^{x}S · S/S = 1 ⟺ ^{x}S · S ∩ Z_{P} (S) = (^{x}S ∩ Z_{P} (S))S = S
⟺ ^{x}S ∩ Z_{P} (S) ⩽ S.

1.13. About extra-special and almost extra-special p-groups.

1.14. Definition. Let p be a prime and P be a finite p-group.

1. The group P is called extra-special if Z(P) = P′ = Φ(P) has order p.

2. The group P is called almost extra-special if P′ = Φ(P) has order p and Z(P) is cyclic of order p^{2}.

Extra-special and almost extra-special p-groups can be classified in the following way. 1.15. Notation. Let H, K and M be groups such that M ⩽ Z(H) and such that there exists an injective map θ : M → Z(K). The central product of H and K with respect to θ will be denoted by H ∗_{θ} K, and simply by H ∗ K if θ is clear from the context.

For any integer r ⩾ 1, we will write H^{∗r} for the central product of r copies of the group H, where M = Z(H), with the convention H^{∗1} = H.

For p ≠ 2, set

M(p) = ⟨x, y | x^{p} = y^{p} = 1, [x^{−1}, y] = [y, x] = ^{y}[y, x]⟩

and

N(p) = ⟨x, y | x^{p^{2}} = y^{p} = 1,^{y}x = x^{1+p}⟩.

1.16. Theorem. Let p be a prime and P be a finite p-group.

1. If P is extra-special, then there exists an integer r ⩾ 1 such that P has order p^{2r+1} and P is isomorphic to only one of the following groups: D_{8}^{∗r} or Q_{8} ∗ D_{8}^{∗(r−1)} if p = 2, and M(p)^{∗r} or N(p) ∗ M(p)^{∗(r−1)} if p ≠ 2.

2. If P is almost extra-special, then there exists an integer r ⩾ 1 such that P has order p^{2r+2} and P is isomorphic to only one of the following groups: D_{8}^{∗r} ∗ C_{4} if p = 2, and M(p)^{∗r} ∗ C_{p^{2}} if p ≠ 2.

Proof. The proof of 1 can be found in Section 5.5 of [14]. As for point 2, one can refer to Sections 2 and 4 of [10].

Observe that if p is odd, the exponent of the group characterizes the isomorphism type of extra-special p-groups, one of them has exponent p and the other one has exponent p^{2}.

If P is an (almost) extra-special group, the quotient P/P′ is elementary abelian, so it can be regarded as a (finite-dimensional) vector space E over the finite field 𝔽_{p}. Moreover, if we take z a generator of P′, then E is endowed with a bilinear form

b : E × E → 𝔽_{p},

that sends an element (u, v) to b(u, v), the element of 𝔽_{p} satisfying [ũ, ṽ] = z^{b(u, v)} for all ũ ∈ u, ṽ ∈ v and u, v ∈ E. This bilinear form is alternating, i.e. b(v, v) = 0 for all v ∈ E, hence it is antisymmetric, i.e. b(u, v) = −b(v, u) for all u, v ∈ E. Section 20 in [11] is concerned with this bilinear form for extra-special groups, but the property of being alternating is called symplectic.

Recall that if f : V ×V → K is a bilinear form on a finite dimensional vector space V over a field K, its left radical V^{⊥} is defined by V^{⊥} = {v ∈ V | f(v, w) = 0 ∀w ∈ V }, and its right radical by ^{⊥}V = {v ∈ V | f(w, v) = 0 ∀w ∈ V }. Clearly V^{⊥} = ^{⊥}V when f is antisymmetric. The rank of f is the codimension of V^{⊥}. The form f is called non-degenerate if V^{⊥} = {0}.

With the help of Lemma 20.4 in [11], we have the following observation.

1.17. Observation. The bilinear form b is non-degenerate if and only if P is extraspecial. If P is almost extra-special, then E^{⊥} = π(Z(P)) is a line in E, where π : P → P/P′ is the projection morphism.

In section 3.2 we will use the following result, which is part of Lemma 2.6 in [9]. For the proof we refer the reader to this reference.

1.18. Lemma. Let P be an (almost) extra-special p-group, and let Q be a non-trivial subgroup of P. Then

1. Q ⊴ P ⇔ P′ ⩽ Q.

2. If Q is not normal in P, then N_{P} (Q) = C_{P} (Q). In particular, it follows that in this case Q is elementary abelian of rank at most r, for the integer r defined as in Theorem 1.16, and we have |Q||C_{P} (Q)| = |P |. Moreover, C_{P} (Q) = Q × U, where U is (almost) extra-special of order |P |/|Q|^{2} or U = Z(P).

2. Cl_{1} of finite p-group algebras for odd p

The goal of this section is to re-write Theorem 9.5 in [17] in terms of genetic bases, in the most possible succinct way. We take the statement of this theorem appearing in Section 1 of [18], which says the following: let p be an odd prime, let P be a finite p-group, and write ℚP ≅ ∏_{i=1}^{k} A_{i}, where A_{i} is simple with irreducible module V_{i} and center K_{i} = End_{ℚP} V_{i}. By Roquette’s Theorem, for each 1 ⩽ i ⩽ k, the field K_{i} is isomorphic to ℚ(ζ_{r_{i}}), where ζ_{r_{i}} is a primitive p^{r_{i}}-th root of unity for some non-negative r_{i}, and A_{i} is isomorphic to a matrix algebra over ℚ(ζ_{r_{i}}).

Consider the abelian group T = ∏_{i=1}^{k}⟨ζ_{r_{i}}⟩. For each h ∈ P, define

ψ_{h} : C_{P} (h) → T, ψ_{h}(g) = (det_{K_{i}}(g, V_{i}^{h}))_{i}

where V_{i}^{h} = {x ∈ V_{i} | hx = x}. Here V_{i}^{h} is viewed as a K_{i}C_{P} (h)-module, so det_{K_{i}}(g, V_{i}^{h}) is the determinant (in K_{i}) of the action of g in V_{i}^{h}. Since P is a pgroup, this determinant is in ⟨ζ_{r_{i}}⟩. Then Theorem 9.5 in [17] can be written in the following way.

2.1. Theorem. Let p be an odd prime and consider T and ψ_{h} : C_{P} (h) → T, for each h ∈ P, as before. Then

Cl_{1}(ℤP) ≅ T/⟨Imψ_{h} | h ∈ P ⟩.

Now, since p is odd, if we let S = {S_{1}, …, S_{k}} be a genetic basis of P, then N_{P} (S_{i})/S_{i} is cyclic for every 1 ⩽ i ⩽ k and each simple ℚP -module V_{i} is isomorphic to V (S_{i}) = Indinf_{N_{P} (S_{i})/S_{i}}^{P}Φ_{N_{P} (S_{i})/S_{i}}. Then the abelian group T defined before

is isomorphic to Γ(P) = ∏_{i=1}^{k} (N_{P} (S_{i})/S_{i}). This is because we can see the module

Φ_{N_{P} (S_{i})/S_{i}} ≅ ℚ(ζ_{r_{i}}), where p^{r_{i}} = |N_{P} (S_{i})/S_{i}| as actually being generated by a generator of N_{P} (S_{i})/S_{i} and thus the action of N_{P} (S_{i})/S_{i} on it can be seen as multiplication on the group. In particular, det_{K_{i}}(g, V_{i}^{h}), which is an element of the field K_{i}, can be regarded as an element in N_{P} (S_{i})/S_{i}. The first step in re-writing Theorem 2.1 is to find this element, for every S_{i}, every element h ∈ P and every g ∈ C_{P} (h).

2.2. Notation. Let p be and odd prime. If V is a simple ℚP -module and S is a genetic subgroup of P corresponding to V, we will write det_{N_{P} (S)/S}(g, V^{h}) for the element in N_{P} (S)/S corresponding to det_{K}(g, V^{h}), where K = End_{ℚP} (V).

2.3. Lemma. Suppose p is an odd prime. Let V be a simple ℚP -module and S be a genetic subgroup of P corresponding to V. Take an element h in P and let H = ⟨h⟩. If g is in C_{P} (H), let [H⟨g⟩\P/N_{P} (S)] be a set of representatives of the double cosets of P on H⟨g⟩ and N_{P} (S). Then we have

det_{N_{P} (S)/S}(g, V^{h}) = ∏_{\begin{matrix}x∈[H⟨g⟩\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{l_{g, x}},

where \overline{l_{g, x}} is determined as follows: the set I_{g,x} = ⟨g⟩ ∩ (H·^{x}N_{P} (S)) is actually a subgroup of ⟨g⟩. Let m = |⟨g⟩ : I_{g,x}|. Then g^{m} can be written as g^{m} = h·^{x}l_{g,x}, for h ∈ H and l_{g,x} ∈ N_{P} (S); the element l_{g, x} may not be unique, but its class \overline{l_{g, x}} in N_{P} (S)/S is, thanks to the conditions on x. Proof. Set Φ for Φ_{N_{P} (S)/S}. Let [P/N_{P} (S)] be a set of representatives of the cosets of P in N_{P} (S). Since V ≅ Indinf_{N_{P} (S)/S}^{P}Φ, we can write it as ⨁_{a∈[P/N_{P} (S)]} a ⊗ Φ. The action of

y ∈ P is given by y(a ⊗ ω) = ya ⊗ ω, which is equal to τ_{y}(a) ⊗ \overline{n_{y, a}}ω, if ya = τ_{y}(a)n_{y, a} for a corresponding n_{y, a} in N_{P} (S), with \overline{n_{y, a}} being its class in N_{P} (S)/S.

If y ∈ H fixes an element u = ∑_{a∈[P/N_{P} (S)]} a ⊗ ω_{a} of V, we have

∑_{a∈[P/N_{P} (S)]} τ_{y}(a) ⊗ \overline{n_{y, a}}ω_{a} = ∑_{a∈[P/N_{P} (S)]} a ⊗ ω_{a} = ∑_{a∈[P/N_{P} (S)]} τ_{y}(a) ⊗ ω_{τ_{y}(a)}.

That is, for every a ∈ [P/N_{P} (S)] we should have \overline{n_{y, a}}ω_{a} = ω_{τ_{y}(a)}. If τ_{y}(a) = a, then we should have that y is in H ∩^{a}N_{P} (S) and that H^{a} ∩ N_{P} (S) ⩽ S, if ω_{a} is different from zero. We consider then the set [H\P/N_{P} (S)] and we have that

u = ∑_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} ∑_{z∈[H/H∩^{x}N_{P} (S)]} zx ⊗ ω_{x}.

This means that a ℚ-basis for V^{h} is given by μ_{x, ω} = ∑_{z∈[H/H∩^{x}N_{P} (S)]} zx ⊗ ω with x

running over F = {x ∈ [H\P/N_{P} (S)] | H^{x} ∩ N_{P} (S) ⩽ S} and ω running over a ℚbasis of Φ. Since Φ ≅ ℚ(ζ_{r}) ≅End_{ℚP} (V), where ζ_{r} is a primitive p^{r}-th root of unity, if the order of N_{P} (S)/S is p^{r}, then a ℚ(ζ_{r})-basis of V^{h} is the set of μ_{x} = μ_{x, 1} with x running over F.

Now, for μ_{x} we have that gμ_{x} is equal to

∑_{z∈[H/H∩^{x}N_{P} (S)]} zh_{g, x}σ_{g}(x) ⊗ \overline{n_{g, x}} 1

if gx = h_{g, x}σ_{g}(x)n_{g, x}. That is

gμ_{x, 1} = h_{g, x}μ_{σ_{g}(x), \overline{n_{g, x}}} = μ_{σ_{g}(x), \overline{n_{g, x}}},

since μ_{σ_{g}(x), \overline{n_{g, x}}} is in V^{h}. So we can write gμ_{x} = \overline{n_{g, x}}μ_{σ_{g}(x)}, with \overline{n_{g, x}} seen as an element in the field ℚ(ζ_{r}). This implies that the action of g in V^{h} is given by a monomial matrix A, the coefficient in the non-zero entry of a row being \overline{n_{g,x}}. Then, the determinant of A is the product of the signature of the permutation σ_{g} by the product of the coefficients \overline{n_{g, x}}. Since p is odd, the signature is +1, and

det(A) = ∏_{\begin{matrix}x∈[H\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{n_{g, x}}.

Observe now that the group H^{x} ∩ N_{P} (S) ⩽ S does not change if we replace x by yx, for y ∈ ⟨g⟩, since g centralizes H. Moreover the intersection I_{g,x} = ⟨g⟩ ∩ (H·^{x}N_{P} (S))

is equal to ⟨g⟩ ∩ (H·(^{x}N_{P} (S) ∩ C_{P} (H⟨g⟩))), by Dedekind’s modular law ([12], Hilfsatz 2.12.c), since g and H centralize H. Hence I_{g,x} is a subgroup of ⟨g⟩. Now a set R of representatives of H\P/N_{P} (S) is the set of elements yx, where x is in a set of representatives of H⟨g⟩\P/N_{P} (S), and y is in a set of representatives of ⟨g⟩/I_{g,x}. The set of elements y can be taken as {1, g, g^{2}, …, g^{m−1}}. For y = g^{i}, with 0 ⩽ i ⩽ m − 2, we have gyx = g^{i+1}x ∈ R, so n_{g,yx} = 1. For i = m−1, we have gyx = g^{m}x = h·x·l_{g,x}.

Our final version of Theorem 2.1 is the following.

2.4. Theorem. Let p be an odd prime and P be a finite p-group. Take a set C of representatives of conjugacy classes of cyclic subgroups of P. For each H ∈ C, let \overline{E}_{H} be a generating set of the factor group C_{P} (H)/H and E_{H} ⊆ C_{P} (H) be a set of representatives of the classes gH ∈ \overline{E}_{H}. Let also S be a genetic basis of P and for each S ∈ S, let [H⟨g⟩\P/N_{P} (S)] be a set of representatives of the double cosets of P on H⟨g⟩ and N_{P} (S). Then

Cl_{1}(ℤP) ≅ (∏_{S∈S} (N_{P} (S)/S)) /R,

where R is the subgroup generated by the elements u_{H,g} = (u_{H,g,S})_{S∈S}, for H ∈ C and g ∈ E_{H}; for S ∈ S, the component u_{H,g,S} of u_{H,g} is given by

(2.5)  u_{H,g,S} = ∏_{\begin{matrix}x∈[H⟨g⟩\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{l_{g, x}},

where \overline{l_{g,x}} is the image in N_{P} (S)/S of the element l_{g,x} ∈ N_{P} (S) determined as follows: let m denote the index of ⟨g⟩ ∩ (H·^{x}N_{P} (S)) in ⟨g⟩. Then g^{m} ∈ H·^{x}N_{P} (S), so g^{m} can be written as g^{m} = h.^{x}l_{g,x}, for some h ∈ H and l_{g,x} ∈ N_{P} (S).

Proof. As we said at the beginning of the section, since p is odd, we have Cl_{1}(ℤP) ≅ Γ(P)/R, where

Γ(P) = ∏_{S∈S} (N_{P} (S)/S)

and R is the subgroup generated by all the elements u_{h,g} = (u_{h,g,S})_{S∈S}, with g ∈ C_{P} (h) and u_{h,g,S} = det_{N_{P} (S)/S} (g, V (S)^{h}), where V (S) = Indinf_{N_{P} (S)/S}^{P}Φ_{N_{P} (S)/S}.

We first observe that u_{h,g} = u_{^{y}h,^{y}g}, for any y ∈ G. Indeed, setting V = V (S), we have a commutative diagram

\xymatrix{V^{h} \ar[r]^{g} \ar[d]_{y} & V^{h} \ar[d]^{y} \\ V^{^{y}h} \ar[r]^{^{y}g} & V^{^{y}h}}

where the arrows are given by the actions of the labelling elements. It follows that the determinant of ^{y}g acting on V^{^{y}h} is equal to the determinant of g acting on V^{h}. Hence to generate the subgroup R of Γ(P), it suffices to take the elements u_{h,g}, where (h, g) runs through a set of representatives of conjugacy classes of pairs of commuting elements in P.

Now clearly for each h ∈ H, the map g ↦ u_{h,g} is a group homomorphism from C_{P} (h) to Γ(P), hence R is generated by the elements u_{h,g}, where h ∈ P and g runs through a set of generators of C_{P} (h). Moreover, as h acts as the identity on V^{h}, the group generated by h is contained in the kernel of this morphism.

Finally, by Lemma 2.3, setting H = ⟨h⟩, we have

u_{h,g,S} = ∏_{\begin{matrix}x∈[H⟨g⟩\P/N_{P} (S)] \\ s.t. H^{x}∩N_{P} (S)⩽S \end{matrix}} \overline{l_{g, x}}

and this depends only on H, so we may denote it by u_{H,g,S}, and by u_{H,g} the corresponding element of Γ(P).

It follows that R is generated by the elements u_{H,g}, where H is a cyclic subgroup of P up to conjugation, and for a given H, the element g runs through a subset of C_{P} (H) which, together with H, generates C_{P} (H). Together with Lemma 2.1, this completes the proof.

To finish the section, we observe that if N is a normal subgroup of a finite p-group P with p odd, then there is surjective deflation morphism

Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)).

2.6. Proposition. Let p be an odd prime and P be a finite p-group. Suppose N is a normal subgroup of P. Let S be a genetic basis of P and S_{N} = {S ∈ S | N ⩽ S}. Let B̃ be the set of subgroups S̃ = S/N of P̃ = P/N, for S ∈ S_{N}.

1. The set {S̃ | S ∈ S_{N}} is a genetic basis of P̃, and for S ∈ S_{N}, the projection P ↦ P̃ induces an isomorphism π_{S} : N_{P} (S)/S → NP̃(S̃)/S̃.

2. The composition

\xymatrix{s : Γ(P) = ∏_{S∈S} (N_{P} (S)/S) \ar[r] & ∏_{S∈S_{N}} (N_{P} (S)/S) \ar[r]^{∏ π_{S}} & ∏_{S̃∈S̃} (NP̃(S̃)/S̃) = Γ(P̃)}

induces a surjective deflation morphism Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)). In particular Cl_{1}(ℤ(P/N)) is isomorphic to a quotient of Cl_{1}(ℤP).

Proof. For Assertion 1, it is clear from the definitions that if S/N is a genetic subgroup of P/N, then S is a genetic subgroup of P. Moreover the irreducible representation of P associated to S is obtained by inflation from P/N to P of the irreducible representation of P/N associated to S/N, up to the obvious isomorphism π_{S} : N_{P} (S)/S ≅ NP̃(S̃)/S̃.

For Assertion 2, as the map s : Γ(P) → Γ(P̃) is surjective, all we have to check is that the subgroup R of defining relations for Cl_{1}(ℤP) ≅ Γ(P)/R is mapped by s inside the corresponding subgroup R̃ of defining relations for Cl_{1}(ℤP̃) ≅ Γ(P̃)/R̃. So let H be a cyclic subgroup of P, let g ∈ C_{P} (H), and let S ∈ S_{N}. Then H̃ = HN/N is cyclic, and g̃ = gN ∈ CP̃(H̃). Moreover the map

H\P/N_{P} (S) ∋ HxN_{P} (S) ↦ H̃x̃NP̃(S̃) ∈ H̃\P̃/NP̃(S̃),

where x̃ = xN ∈ P̃, is a bijection, since HxN_{P} (S) = HxNN_{P} (S) = HNxN_{P} (S) as N_{P} (S) ⩾ S ⩾ N. Hence we may identify the sets of representatives [H\P/N_{P} (S)] and [H̃\P̃/NP̃(S̃)] via this map. Moreover

H^{x} ∩ N_{P} (S) ⩽ S ⟺ (HN)^{x} ∩ N_{P} (S) ⩽ S ⟺ H̃x̃ ∩ NP̃(S̃) ⩽ S̃.

Now, for x ∈ [H\P/N_{P} (S)] such that H^{x}∩N_{P} (S) ⩽ S, the equality gx = h_{g,x}σ_{g}(x)n_{g,x},

where h_{g,x} ∈ H, σ_{g}(x) ∈ [H\P/N_{P} (S)], and n_{g,x} ∈ N_{P} (S), implies g̃x̃ = h̃_{g,x}{ \atop σ_{g}(x)}ñ_{g,x}

in P/N. In other words π_{S}(\overline{n_{g,x}}) = \overline{n_{g̃,x̃}}, that is s(u_{H,g}) = u_{H̃,g̃}. This completes the proof.

2.7. Remark. It is shown in [4] (where Γ(P) is called the genome of P) that the map s is the deflation map in a structure of biset functor on Γ, but we will not need the corresponding additional operations of induction, restriction and inflation in this paper. It also follows from [4] that the deflation map Def_{P/N}^{P} : Cl_{1}(ℤP) → Cl_{1}(ℤ(P/N)) we obtain here is the same as the map given by Corollary 3.10 of [17].

3. Computing some Whitehead groups

As we noted in Section 1, the examples we will consider in this section are all finite p-groups with p odd that satisfy that the group SK_{1}(ℤP) is equal to Cl_{1}(ℤP). Hence, we will use Theorem 2.4 to calculate Cl_{1}(ℤP). If P is abelian, Theorem 2.4 has a simpler expression, as it was already noted in Observation 1.13 of [18].

As we said in the introduction, to calculate the free rank of the Whitehead group of the groups in question, we will use Theorem 2.6 in [17]. We will also use Exercise 13.9 in Serre [20], which says that if G is a group of odd order and c is the number of irreducible non-isomorphic complex representations of G, then (c + 1)/2 is the number of irreducible non-isomorphic real representations of G.

We introduce some notation that will be helpful in both of our examples. 3.1. Notation. Let p be a prime. Suppose that W is a finite-dimensional vector space over the finite field 𝔽_{p}. We denote by S(W) the symmetric algebra of W and by S^{p}(W) its homogeneous part of degree p. If ψ : W → 𝔽_{p} is a linear functional, the map

w_{1} ⊗_{𝔽_{p}} … ⊗_{𝔽_{p}} w_{n} ∈ W^{⊗n} ↦ ψ(w_{1}) … ψ(w_{n}) ∈ 𝔽_{p}

induces a well defined linear functional S(W) → 𝔽_{p}, that we denote by A ↦ A(ψ).

The choice of a basis {x_{1}, …, x_{k}} of W over 𝔽_{p} yields a standard identification of S(W) with the polynomial ring 𝔽_{p}[x_{1}, …, x_{k}].

With such an identification, if A = A(x_{1}, …, x_{k}) ∈ S(W) and ψ is a linear form on W, then A(ψ) = A(ψ(x_{1}), …, ψ(x_{k})). In particular A(ψ) = 0 for all ψ if and only if the polynomial function associated to A is equal to zero, that is if A(r_{1}, …, r_{k}) = 0 for any (r_{1}, …, r_{k}) ∈ 𝔽_{p}^{k}: indeed since {x_{1}, …, x_{k}} is a basis of W, for any such ktuple (r_{1}, …, r_{k}) ∈ 𝔽_{p}^{k}, there is a unique linear form ψ on W such that ψ(x_{i}) = r_{i} for 1 ⩽ i ⩽ k.

3.2. Elementary abelian p-groups.

3.3. Lemma. Let p be an odd prime and P be an elementary abelian p-group of rank k, the free rank of W h(P) is equal to

\frac{(p^{k} − 1)(p − 3)}{2(p − 1)}.

Proof. The number of non-isomorphic irreducible ℝ-representations of P is (p^{k} +1)/2. On the other hand, the number of non-isomorphic irreducible ℚ-representations of P is equal to (p^{k} +p−2)/(p−1), since the genetic basis for P is given by all its subgroups of index p plus P itself. The result follows from Theorem 2.6 in [17].

The description of SK_{1} for elementary abelian groups appeared first in Alperin et al. [1]. We prove this result using our combinatorial approach with genetic bases. Our proof has some similarities with the one in [1], but makes no use of characters. It will also be useful when dealing with extra-special p-groups.

3.4. Lemma. Let p be a prime and let W be a finite-dimensional vector space over 𝔽_{p}. For x and y in W, we set B_{x, y} = x^{p−1}y ∈ S^{p}(W). Then S^{p}(W) is generated by the elements B_{x, y} with x and y running over W. Proof. Recall that if x_{1}, …, x_{m} are m (not necessarily different) commuting variables, then for any n

(x_{1} + ⋯ + x_{m})^{n} = ∑_{\begin{matrix}α_{1},...,α_{m} s.t. \\ α_{1}+···+α_{m}=n \end{matrix}} \frac{n!}{α_{1}! ⋯ α_{m}!}x_{1}^{α_{1}} ⋯ x_{m}^{α_{m}}.

This allows us to show that for any n

(3.5)  ∑_{∅≠A⊆{1,...,n}}(−1)^{n−|A|} (∑_{i∈A} x_{i})^{n} = n! x_{1} ⋯ x_{n},

and so if p is prime, then

∑_{∅≠A⊆{1,...,p−1}}(−1)^{p−1−|A|} (∑_{i∈A} x_{i})^{p−1} x_{p} = (p − 1)! x_{1} ⋯ x_{p},

which by Wilson’s Lemma gives

∑_{∅≠A⊆{1,...,p−1}}(−1)^{p−|A|} (∑_{i∈A} x_{i})^{p−1} x_{p} = x_{1} ⋯ x_{p},

that is

(3.6)  ∑_{∅≠A⊆{1,...,p−1}}(−1)^{p−|A|}B_{∑_{i∈A} x_{i},x_{p}} = x_{1} ⋯ x_{p}.

This completes the proof, by taking x_{1}, …, x_{p} ∈ W = S^{1}(W), which indeed commute in S(W).

3.7. Theorem. [Theorem 2.4 in [1]] Let p be an odd prime and P be an elementary abelian p-group of rank k, then SK_{1}(ℤP) is isomorphic to (C_{p})^{N} where

N = \frac{p^{k} − 1}{p − 1} − \binom{p + k − 1}{p}.

Proof. As we said before, the genetic basis of P consists of P itself and all its subgroups of index p, so SK_{1}(ℤP) is isomorphic to the quotient of

Γ(P) = ∏_{[P:Q]=p}(P/Q)

by the subgroup generated by the elements u_{x, y} for x, y in P, where

(u_{x, y})_{Q} = \{ \begin{matrix}yQ & if x ∈ Q \\ 1 & otherwise. \end{matrix}

On the other hand, we can see P as a vector space over 𝔽_{p}, and for each subgroup Q of index p, i.e. for each hyperplane Q of P, consider ψ_{Q} : P → 𝔽_{p}, a linear functional with kernel Q. The product of these ψ_{Q} induces an isomorphism from Γ(P) to

V = ∏_{[P:Q]=p} 𝔽_{p},

and the elements u_{x, y} can be seen as

(u_{x, y})_{Q} = \{ \begin{matrix}ψ_{Q}(y) & if x ∈ Q \\ 0 & otherwise. \end{matrix}

We define a morphism r : S^{p}(P) → V, sending A ∈ S^{p}(P) to the vector whose Qcomponent is equal to A(ψ_{Q}). We will show that Im(r) is equal to the subspace of V generated by the elements u_{x, y}, and that r is injective. This will give us the result.

We show first that the elements u_{x, y} are in Im(r). For x, y ∈ P, let B_{x, y} = x^{p−1}y ∈ S^{p}(P). Then

B_{x, y}(ψ_{Q}) = ψ_{Q}(x)^{p−1}ψ_{Q}(y) = \{ \begin{matrix}0 & if x ∈ Q \\ ψ_{Q}(y) & otherwise \end{matrix}

since λ^{p−1} = 1 if λ is in 𝔽_{p} − {0} and 0^{p−1} = 0. In particular B_{y,y}(ψ_{Q}) = ψ_{Q}(y) for all y ∈ P, thus r(B_{y,y} − B_{x, y}) = u_{x, y}.

On the other hand, by Lemma 3.4, S^{p}(P) is generated by the elements B_{x, y} where x and y run through P, so we have that Im(r) is contained in (hence equal to) the subspace of V generated by the elements u_{x, y}.

Finally, we prove that r is injective. Let A be in the kernel of r, then A(ψ_{Q}) = 0 for every Q of index p in P. If ψ is any other linear functional of P with kernel Q, then there exists λ ∈ 𝔽_{p} such that ψ = λψ_{Q}, and so A(ψ) = λ^{p}A(ψ_{Q}) = 0, since A is homogeneous of degree p. Choosing a basis of P over 𝔽_{p} as in Notation 3.1, we can view A as a homogeneous polynomial of degree p, and the polynomial function A is zero. It remains to see that A is actually the zero polynomial, but this follows from Lemma 2.1 of [1].

3.8. Corollary. Let p be an odd prime, and P be a finite p-group. If |P/Φ(P)| = p^{k}, then SK_{1}(ℤP) has a subquotient isomorphic to (C_{p})^{N}, where

N = \frac{p^{k} − 1}{p − 1} − \binom{p + k − 1}{p},

and in particular SK_{1}(ℤP) ≠ 0 if k ⩾ 3.

Proof. Indeed Cl_{1}(ℤ(P/Φ(P))) ≅ (C_{p})^{N}, for N = \frac{p^{k}−1}{p−1} − \binom{p+k−1}{p}, by Theorem 3.7. Moreover, this group is a quotient of Cl_{1}(ℤP), by Proposition 2.6. Finally Cl_{1}(ℤP) is a subgroup of SK_{1}(ℤP), and N > 0 if k ⩾ 3.

3.9. Extra-special and almost extra-special p-groups.

We begin by finding a genetic basis of an (almost) extra-special p-group.

3.10. Proposition. Let p be a prime and P be an (almost) extra-special p-group. A genetic basis of P is given by all its subgroups of index p, together with P and a subgroup Y of maximal order such that Y ∩ Z(P) = 1. In particular P has a unique faithful rational irreducible representation, up to isomorphism. Proof. We abbreviate Z(P) by Z.

By theorems 1.7 and 1.8, the subgroups of P of index 1 or p are genetic and are not linked modulo _{P} . They clearly intersect Z non-trivially. On the other hand, any genetic subgroup S ≠ P which intersects Z non-trivially must have index p, since P′ ⩽ S and so the cyclic group P/S ≅ (P/P′)/(S/P′), should have order p. This implies that if there is another group in S, it must intersect Z trivially.

Let Y ⩽ P be of maximal order with the property Y ∩ Z = 1. By Lemma 1.18, we have that C_{P} (Y) = N_{P} (Y) = Y Z. In particular N_{P} (Y) ⊴ P and N_{P} (Y)/Y ≅ Z is a Roquette group. Then Y is genetic, by Remark 1.12. Also, by Lemma 1.18, we have that if Y_{1} ⩽ P is a group such that Y_{1} ∩ Z = 1, but it is not maximal order with this property, then C_{P} (Y_{1}) = N_{P} (Y_{1}), but N_{P} (Y_{1})/Y_{1} is not cyclic. Thus Y_{1} is not a genetic subgroup of P.

Finally, if Y is a subgroup of P of maximal order such that Y ∩ Z = 1, then Y has |P/Y Z| = |Y | distinct conjugates in P, by Lemma 1.18. These conjugates are subgroups of index p in the elementary abelian group Y P′, and they all intersect trivially (that is, they don’t contain) the group P′. Since there are exactly |Y | subgroups of Y P′ not containing P′, these subgroups are exactly the conjugates of Y in P.

Now if Y_{0} is another subgroup of P such that Y_{0} ∩ Z = 1, then Y_{0} ∩ Y P′ is a subgroup of Y P′ which does not contain P′. Hence it is contained in some conjugate of Y, and there exists x ∈ P such that Y_{0} ∩ Y P′ ⩽ Y^{x}. It follows that Y_{0} ∩ Y Z ⩽ Y^{x}, for Y P′ is the subgroup of Y Z consisting of elements of order at most p. In other words

xY_{0} ∩ Z_{P} (Y) = ^{x}Y_{0} ∩ Y Z = ^{x}(Y_{0} ∩ Y Z) ⩽ Y.

Now if Y_{0} is another subgroup of maximal order such that Y_{0} ∩ Z = 1, exchanging the roles of Y and Y_{0} in the previous argument shows that there also exists an element y ∈ P such that ^{y}Y ∩ Y_{0}Z ⩽ Y_{0}. By Theorem 1.8, it follows that Y_{0} _{P} Y. The last assertion now follows from Lemma 1.10.

As a first consequence of this result we have.

3.11. Lemma. Let p be an odd prime, and n be a positive integer.

1. Let P be an extra-special p-group of order p^{2n+1}. Then the free rank of W h(P) is equal to

\frac{(p^{2n} + p − 2)(p − 3)}{2(p − 1)}.

2. Let P be an almost extra-special group of order p^{2n+2}. Then the free rank of W h(P) is equal to

\frac{(p^{2n+1} + p^{2} + p + 1)(p − 3) + 8}{2(p − 1)}.

Proof. The free rank of W h(P) is equal to r − q, where r (resp. q) is the number of irreducible real (resp. rational) representations of P, up to isomorphism. In general, for a finite p-group P and a field K of characteristic 0, the irreducible representations of P over K can be recovered from the knowledge of a genetic basis B of P (see [2]): this is because the functor R_{K} of representations of p-groups over K is rational in the sense of Definition 10.1.3 of [7], as can be easily deduced from Theorem 10.6.1 of [7]. A proof of this fact can also be found in [8]. In particular, the number l_{K}(P) of such representations, up to isomorphism, is equal to

l_{K}(P) = ∑_{S∈B} ∂l_{K}(N_{P} (S)/S),

where ∂l_{K}(Q) denotes the number of faithful irreducible representations of a p-group Q over K, up to isomorphism. For a Roquette p-group Q, we have moreover ∂l_{K}(Q) = 1 if Q = 1, and ∂l_{K}(Q) = l_{K}(Q) − l_{K}(Q/Z) otherwise, where Z is the unique central subgroup of order p of Q.

If p is odd, all the groups N_{P} (S)/S, for S ∈ B, are cyclic. Now for Q = C_{p^{m}}, with m ⩾ 0, we have

l_{ℝ}(Q) = \frac{p^{m} + 1}{2} and l_{ℚ}(Q) = m + 1.

It follows that ∂l_{ℝ}(Q) = \frac{p^{m}−p^{m−1}}{2} if m > 0, and ∂l_{ℝ}(Q) = 1 if m = 0. On the other hand ∂l_{ℚ}(Q) = 1 for any m.

In case P is extra-special of order p^{2n+1}, the genetic basis obtained in Proposition 3.10 consists of the group S = P, for which N_{P} (S)/S is trivial, of \frac{p^{2n}−1}{p−1} subgroups S of index p in P, for which N_{P} (S)/S ≅ C_{p}, and the subgroup S = Y, for which N_{P} (S)/S ≅ Z(P) ≅ C_{p}. This gives

r = l_{ℝ}(P) = 1 + \frac{p^{2n} − 1}{p − 1} \frac{p − 1}{2} + \frac{p − 1}{2} = \frac{p^{2n} + p}{2},

and

q = l_{ℚ}(P) = 1 + \frac{p^{2n} − 1}{p − 1} + 1 = \frac{p^{2n} + 2p − 3}{p − 1}.

In case P is almost extra-special of order p^{2n+2}, the genetic basis obtained in Proposition 3.10 consists of the group S = P, for which N_{P} (S)/S is trivial, of \frac{p^{2n+1}−1}{p−1} subgroups S of index p in P, for which N_{P} (S)/S ≅ C_{p}, and the subgroup S = Y, for which N_{P} (S)/S ≅ Z(P) ≅ C_{p^{2}}. This gives

r = l_{ℝ}(P) = 1 + \frac{p^{2n+1} − 1}{p − 1} \frac{p − 1}{2} + \frac{p^{2} − p}{2} = \frac{p^{2n+1} + p^{2} − p + 1}{2},

and

q = l_{ℚ}(P) = 1 + \frac{p^{2n+1} − 1}{p − 1} + 1 = \frac{p^{2n+1} + 2p − 3}{p − 1}.

This completes the proof.

To calculate Cl_{1}(ℤP) we will need the following result.

3.12. Lemma. Let p be an odd prime. Let W be a vector space over 𝔽_{p} of finite dimension k, which is endowed with a bilinear, alternating form b : W × W → 𝔽_{p}. Suppose that the rank of b is not equal to 2.

For x and y in W, we still set B_{x, y} = x^{p−1}y ∈ S^{p}(W). Then we have

S^{p}(W) = ⟨B_{x, y} | x, y ∈ W s.t. b(x, y) = 0⟩.

Proof. We will write O for ⟨B_{x, y} | x, y ∈ W s.t. b(x, y) = 0⟩. We will prove that B_{x,y} ∈ O for any x, y ∈ W, and by Lemma 3.4, it will follow that S^{p}(W) = O.

Observe that, since B_{x,y} ∈ O for an x, y ∈ W with b(x, y) = 0, it follows from formula 3.6 that x_{1}x_{2} … x_{p−1}y ∈ O for any elements x_{1}, …, x_{p−1} of W such that b(x_{i}, y) = 0 for 1 ⩽ i ⩽ p − 1.

If b = 0, there is nothing to prove. Otherwise let x, y ∈ W such that b(x, y) ≠ 0. Since B_{x,λy} = λB_{x,y} for λ ∈ 𝔽_{p}, to prove that B_{x,y} ∈ O, we can assume without loss of generality that b(x, y) = 1, up to replacing y by a suitable scalar multiple. If the restriction of b to ⟨x, y⟩^{⊥} was identically 0, then ⟨x, y⟩^{⊥} would be precisely the radical of b, so b would have rank 2, contradicting our assumption. Hence we can find z, t ∈ ⟨x, y⟩^{⊥} such that b(z, t) ≠ 0, and up to replacing t by some scalar multiple, we can assume that b(z, t) = 1.

Now let α ∈ 𝔽_{p}, and set u = αx + t and v = y + αz. Then

b(u, v) = αb(x, y) + α^{2}b(x, z) + b(t, y) + αb(t, z) = 0,

since b(x, y) = 1 = −b(t, z) and b(x, z) = 0 = b(t, y).

It follows that B_{u,v} ∈ O. But B_{u,v} is equal to

(3.13) (αx + t)^{p−1}(y + αz) = ∑_{i=0}^{p−1} \binom{p − 1}{i}α^{i}x^{i}yt^{p−1−i} + ∑_{i=0}^{p−1} \binom{p − 1}{i}α^{i+1}t^{p−1−i}zx^{i}.

By the observation at the beginning of the proof, the element x^{i}yt^{p−1−i} is in O whenever p − 1 − i > 0, since b(x, t) = b(y, t) = b(t, t) = 0. Similarly t^{p−1−i}zx^{i} ∈ O if i > 0, since b(t, x) = b(z, x) = b(x, x) = 0. It follows that in (3.13), the only elements possibly not in O correspond to p − 1 − i = 0 in the first summation and to i = 0 in the second. Hence

α^{p−1}x^{p−1}y + αt^{p−1}z ∈ O,

and this holds for any α ∈ 𝔽_{p}. For α = 1, this gives x^{p−1}y +t^{p−1}z ∈ O, and for α = −1, this gives x^{p−1}y − t^{p−1}z ∈ O. It follows that x^{p−1}y ∈ O, as was to be shown.

3.14. Remark. If the rank of b is equal to 2, then the result of Lemma 3.12 is no longer true: for example in the non-degenerate case, that is when W has dimension 2, saying that b(x, y) = 0 for x ≠ 0 is equivalent to saying that y is a scalar multiple of x. In this case O is the subspace of S^{p}(W) generated by the elements x^{p}, for x ∈ W. So O has dimension 2, whereas S^{p}(W) has dimension p + 1.

We now come to our main theorem, describing the structure of Cl_{1}(ℤP) when P is an extra-special or almost extra-special p-group for p odd. We first recall that Oliver ([17] Example 7 page 16) showed that if P is extra-special of order p^{3}, then Cl_{1}(ℤP) ≅ (C_{p})^{p−1}, and that if P is almost extra-special of order p^{4}, then Cl_{1}(ℤP) ≅ (C_{p})^{(p^{2}+p−2)/2}. Hence in what follows, we may assume that P is an extra-special group of order at least p^{5}, or an almost extra-special p-group of order at least p^{6}.

3.15. Notation. Let p be an odd prime and n be a positive integer. Let P be an extra-special p-group of order p^{2n+1} or an almost extra-special p-group of order p^{2n+2}. Let Z denote the center of P, and N = P′ ⩽ Z be the Frattini subgroup of P. Let Y be a subgroup of P of maximal order such that Y ∩ Z = 1, as in Proposition 3.10. Recall that Y is elementary abelian. In any case, the group P can be written as a semidirect product P = X · Y, where X ⩾ Z is an abelian normal subgroup of P with C_{P} (X) = X and Y ∩ X = 1:

• If P is extra-special of exponent p, the group X is equal to C × X_{0}, for some subgroup X_{0} ≅ (C_{p})^{n} and C = N = Z.

• If P is extra-special of exponent p^{2}, the group X is equal to C × X_{0}, for some subgroup X_{0} ≅ (C_{p})^{n−1}, some subgroup C ≅ C_{p^{2}}, and N = Z < C.

• If P is almost extra-special, then X = C × X_{0}, for some subgroup X_{0} ≅ (C_{p})^{n}, and N < Z = C ≅ C_{p^{2}}.

So in all cases we have X = C × X_{0}, for some cyclic subgroup C ⩾ Z ⩾ N.

Moreover the subgroup Y is elementary abelian of order p^{n}. It is maximal subject to the condition Y ∩ Z = 1, so by Proposition 3.10, we have a genetic basis of P consisting of P itself, its subgroups of index p, and Y. The normalizer N_{P} (Y) is equal to Z · Y, so

Γ(P) ≅ ( ∏_{[P:Q]=p}(P/Q)) × Z.

3.16. Theorem. Let p be an odd prime, and let P be an extra-special p-group of order at least p^{5}, or an almost extra-special p-group of order at least p^{6}. Let N = P′ be the Frattini subgroup of P, and Z be the center of P. Then there is a split sequence of abelian groups

\xymatrix{0 \ar[r] & K \ar[r] & Cl_{1}(ℤP) \ar[r]^{^{Def{P \atop P/N}}} & Cl_{1}(ℤ(P/N)) \ar[r] & 0,}

where K is cyclic, isomorphic to a quotient of Z. In particular Cl_{1}(ℤP) is isomorphic to K × (C_{p})^{M}, where K is cyclic of order dividing p^{2}, and M = \frac{p^{k−1}−1}{p−1} − \binom{p+k−2}{p} if |P | = p^{k}. Proof. The product ∏_{[P:Q]=p}(P/Q) identifies with Γ(P/N), and we have a surjective

projection map Def_{P/N}^{P} : Γ(P) → Γ(P/N), with kernel isomorphic to Z. By Proposition 2.6, this map induces a surjective deflation map

Def_{P/N}^{P} : Cl_{1}(ℤP) = Γ(P)/R → Cl_{1}(ℤ(P/N)) = Γ(P/N)/\overline{R},

where R is the subgroup of Γ(P) generated by the elements u_{H,g} introduced in Theorem 2.4, and H is a cyclic subgroup of P with g ∈ C_{P} (H). Similarly \overline{R} is the corresponding subgroup of Γ(P/N) generated by the elements u_{F,c}, where F is a cyclic subgroup of P/N and c ∈ P/N (we always have c ∈ C_{P/N}(F), as P/N is abelian).

The proof of Theorem 2.6 shows that Def_{P/N}^{P}(u_{H,g}) = u_{HN/N,gN}. Conversely, if F is a cyclic subgroup of P/N, generated by f, and if c ∈ P/N, then there exists a pair (H, g) of a cyclic subgroup H of P and an element g ∈ C_{P} (H) such that HN/N = F and gN = c if and only if b(f, c) = 0, where b is the bilinear alternating form on P/N with values in 𝔽_{p} induced by taking commutators in P. Our assumptions on P imply that the rank of b is not 2, so we can apply Lemma 3.12. This shows that the subspace of Γ(P/N) generated by the elements u_{F,c}, where F = ⟨f⟩ for f ∈ P/N, and c ∈ P/N such that b(f, c) = 0, generate S^{p}(P/N) = \overline{R}, by Lemma 3.4. It follows that Def_{P/N}^{P} induces a surjective map e : R → \overline{R}. Let L denote the kernel of this map. We have a commutative diagram with exact rows

\xymatrix{0 \ar[r] & L \ar[d]^{l} \ar[r] & R \ar[d]^{i} \ar[r]^{e} & \overline{R} \ar[d]^{j} \ar[r] & 0 \\ 0 \ar[r] & Z \ar[r] & Γ(P) \ar[r]^{_{Def{P \atop P/N}}} & Γ(P/N) \ar[r] & 0}

where the vertical maps i and j are the inclusion maps. The Snake’s Lemma now shows that the map l is injective, and moreover we have an exact sequence of cokernels

(3.17)  \xymatrix{0 \ar[r] & K \ar[r] & Cl_{1}(ℤP) \ar[r]^{^{Def{P \atop P/N}}} & Cl_{1}(ℤ(P/N)) \ar[r] & 0,}

where K = Z/l(L). Since the kernel Z of Def_{P/N}^{P} corresponds to the component of Γ(P) indexed by Y, the image l(L) is generated by the components w_{Y} ∈ N_{P} (Y)/Y ≅ Z of vectors w in the kernel of the deflation map e : R → \overline{R}. Hence l(L) is a subgroup of the group generated by all the elements u_{H,g,Y}, where H is a cyclic subgroup of P and g ∈ C_{P} (H).

It remains to see that the exact sequence 3.17 splits. To see this, consider the completed diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & L \ar[d]^{l} \ar[r] & R \ar[d]^{i} \ar[r]^{e} & \overline{R} \ar[d]^{j} \ar[r] & 0 \\ 0 \ar[r] & Z \ar[d] \ar[r] & Γ(P) \ar[d]^{a} \ar[r]^{c} & Γ(P/N) \ar@{-->}[l]^{t} \ar[d]^{b} \ar[r] & 0 \\ 0 \ar[r] & K \ar[d] \ar[r] & Cl_{1}(ℤP) \ar[d] \ar[r]^{d} & Cl_{1}(ℤ(P/N)) \ar@{-->}[u]^{s} \ar[d] \ar[r] & 0, \\  & 0 & 0 & 0}

where a and b are the projection maps, and c, d are the respective deflation maps, and e is the restriction of c to R. The map b is split surjective, because Γ(P/N) and Cl_{1}(ℤ(P/N)) are both elementary abelian. Similarly, the map c is split surjective by construction. Let s be a section of b and let t : Γ(P/N) → Γ(P) be a section of c. Then

d ∘ a ∘ t ∘ s = b ∘ c ∘ t ∘ s = b ∘ s = Id,

so the map a ∘ t ∘ s is a section of d. To complete the proof, observe that K is a isomorphic to a quotient of Z, and that Z is cyclic of order p or p^{2}.

3.18. Remark. It follows from this proof that to obtain a minimal set M of generators of Cl_{1}(ℤP), we can take a subset (v_{S})_{S∈S} of the canonical basis of Γ(P/N) (as 𝔽_{p}-vector space), corresponding to a set S of subgroups of index p of P, which maps by b to a basis of Cl_{1}(ℤ(P/N)). Then v_{S} is a generator of the component P/S ≅ (P/N)/(S/N) of Γ(P), for S ∈ S. If K is trivial^{3}, we set T = S. Otherwise, we also choose a generator v_{Y} of Z ≅ N_{P} (Y)/Y, and we set T = S ⊔ {Y }. Then we take for M the image in Cl_{1}(ℤP) by the projection map a of the set {v_{S} | S ∈ T }.

3This will occur only in Assertion 1 of Proposition 3.22 below, i.e. when P is extra-special of order p^{5} and exponent p^{2}

As we can see from this procedure, the splitting of the exact sequence of Theorem 3.16 is not canonical: first we have made a choice for the genetic subgroup Y, and then we also make a choice of a set S and of generators v_{S}, for S ∈ T .

We now come to the proof of Theorem A, which we split in two parts: first the “generic” case, when the group P is large enough, and then two special cases of “small” groups of order p^{5} and p^{6}. The generic case is the following one.

3.19. Theorem. Let p be an odd prime, and let P be a p-group of order p^{k} which is either:

1. extra-special of exponent p with k ⩾ 5, or

2. extra-special of exponent p^{2} with k ⩾ 7, or

3. almost extra-special with k ⩾ 8.

Then Cl_{1}(ℤP) is (non-canonically) isomorphic to Z × (C_{p})^{M}, where Z is the center of P (so Z ≅ C_{p} in cases 1 and 2, and Z ≅ C_{p^{2}} in case 3), and M = \frac{p^{k−1}−1}{p−1} − \binom{p+k−2}{p}. Proof. We keep Notation 3.15 throughout. We know from Theorem 3.16 that Cl_{1}(ℤP) ≅ K × (C_{p})^{M}, where M = \frac{p^{k−1}−1}{p−1} − \binom{p+k−2}{p}.

Moreover the group K is isomorphic to the quotient of Z by the subgroup generated by the elements w_{Y}, where w is an element in the kernel of the deflation map e : R → \overline{R}. Such an element w_{Y} is a product of elements u_{H,g,Y}, for some pairs (H, g) of a cyclic subgroup H of P, and an element g ∈ C_{P} (H). We start by computing u_{H,g,Y} for such a pair (H, g), using formula 2.5. We will show that u_{H,g,Y} = 1 in the cases of the statement of the theorem, and it will follow that K ≅ Z. We have

u_{H,g,Y} = ∏_{x∈D} \overline{l_{g, x}}

where D is a chosen set of representatives of those double cosets H⟨g⟩xN_{P} (Y) in P for which H^{x} ∩ N_{P} (Y) ⩽ Y. For each x ∈ D, if m_{x} is the index of ⟨g⟩ ∩ (H ·^{x}N_{P} (Y)) in ⟨g⟩, the element g^{m_{x}} can be written g^{m_{x}} = h_{x} ·^{x}l_{g,x}, for h_{x} ∈ H and l_{g,x} ∈ N_{P} (Y). Since N_{P} (Y) = ZY is a normal subgroup of p, by Lemma 1.18, the group H ·^{x}N_{P} (Y) and the integer m_{x} do not depend on x. Let m denote this integer. Since g^{p} ∈ Z for any g ∈ P, we have m = 1 if g ∈ HZY, and m = p otherwise.

Since H⟨g⟩ZY is also a normal subgroup of P, we have H⟨g⟩tZY = H⟨g⟩ZY t = tH⟨g⟩ZY, for any t ∈ P. Hence D is a subset of a set of representatives of P/H⟨g⟩ZY. If D is empty, we have u_{H,g,Y} = 1, so we can assume D ≠ ∅. If H^{t} ∩ ZY ⩽ Y, then H^{t} ∩ Z ⩽ Y ∩ Z = 1. Then H ∩ Z = 1, so H has order 1 or p, since h^{p} ∈ Z for any h ∈ P.

• If H ∩ ZY = 1 then H^{t} ∩ ZY = (H ∩ ZY)^{t} = 1 for any t ∈ P, so our set D is a set of representatives of P/H⟨g⟩ZY.

• If H ∩ZY ≠ 1, then H ⩽ ZY. If there exists t_{0} ∈ P with H^{t_{0}} ∩ZY ⩽ Y, we have H^{t_{0}} ⩽ Y, and up to replacing H by H^{t_{0}}, we can assume H ⩽ Y. Then for t ∈ P, we have H^{t} ∩ ZY = H^{t}, and H^{t} ⩽ Y implies [H, H^{t}] ⩽ Z ∩ Y = 1. In other words H^{t} ∩ ZY ⩽ Y if and only if t ∈ C_{P} (H). Here D is a set of representatives of C_{P} (H)/H⟨g⟩ZY = C_{P} (H)/⟨g⟩ZY.

So in each case, there are subgroups M and Q of P with ZY ⩽ M ⩽ Q such that D is a set of representatives of Q/M.

We have g^{m} = h·z·y, for some h ∈ H, z ∈ Z and y ∈ Y. Observe that z = g^{p} and h = y = 1 if m = p. Then for x ∈ D, we have g^{m} = h·z·^{x}(y^{x}) = h·^{x}(z[y, x]y), so with the identification N_{P} (Y)/Y ≅ Z, we have l_{g,x} = z[y, x]. It follows that u_{H,g,Y} = ∏_{x∈D} (z[y, x]) = z^{|D|}[y, ∏_{x∈D} x]. Now the commutator [y, ∏_{x∈D} x] only depends of the images

of y and ∏_{x∈D} x in the elementary abelian group P/N. As M ⩾ N, the map x ↦ \overline{x} = xN

is a bijection from Q/M to \overline{Q}/\overline{M}, where \overline{Q} = Q/N and \overline{M} = M/N. The subgroup \overline{M} admits a supplement W in the elementary abelian group \overline{Q}, and W is a set of representatives of \overline{Q}/\overline{M}. Since ∑_{w∈W} w = 0 as p is odd, it follows that ∏_{x∈D} x maps to 0

in P/N, that is ∏_{x∈D} x ∈ N ⩽ Z, and [y, ∏_{x∈D} x] = 1. This gives finally

(3.20)  u_{H,g,Y} = \{ \begin{matrix}z^{|D|} & if g ∈ HzY for some z ∈ Z, \\ g^{p|D|} & otherwise. \end{matrix}

• If P is extra-special of order p^{2l+1}, with l ⩾ 2, then Y has order p^{l} and Z has order p.

- As we said, if g ∈ HzY, for z ∈ Z, we have u_{H,g,Y} = z^{|D|}. If H ≰ ZY, then

|D| = |P : H⟨g⟩ZY | = |P : HZY | ⩾ p^{2l+1}/(p·p·p^{l}) = p^{l−1}.

Now if we assume H ⩽ Y, then g ∈ HZY = ZY, and

|D| = |C_{P} (H) : ⟨g⟩ZY | = [C_{P} (H) : ZY | ⩾ p^{2l}/p^{l+1} = p^{l−1}

again. For l ⩾ 2, this is a multiple of p = |Z|, and u_{H,g,Y} = 1.

- If g ∉ HZY, then u_{H,g,Y} = g^{p|D|}. So u_{H,g,Y} = 1 unless g has order p^{2}, and then g^{p} generates Z. If H ≰ ZY, then

|D| = |P : H⟨g⟩ZY | = |P : H⟨g⟩Y | ⩾ p^{2l+1}/p·p^{2}·p^{l} = p^{l−2}.

On the other hand, if H ⩽ Y, then

|D| = |C_{P} (H) : ⟨g⟩ZY | = |C_{P} (H) : ⟨g⟩Y | ⩾ p^{2l}/p^{2}·p^{l} = p^{l−2}

again. Hence |D| is a multiple of p if l ⩾ 3, so u_{H,g,Y} = 1.

So u_{H,g,Y} = 1 if P is extra-special, unless possibly if P has exponent p^{2} and order p^{5}.

• If P is almost extra-special of order p^{2l+2}, with l ⩾ 2, then Y has order p^{l} and Z is cyclic of order p^{2}.

- If g ∈ HzY, for z ∈ Z, then u_{H,g,Y} = z^{|D|}. If H ≰ ZY, we have

|D| = |P : H⟨g⟩ZY | = |P : HZY | ⩾ p^{2l+2}/(p·p^{2}·p^{l}) = p^{l−1}.

If we now assume H ⩽ Y, then g ∈ HZY = ZY, and

|D| = |C_{P} (H) : ⟨g⟩ZY | = |C_{P} (H) : ZY | ⩾ p^{2l+1}/p^{2}·p^{l} = p^{l−1}

again. Hence then |D| is a multiple of p^{2} if l ⩾ 3, so u_{H,g,Y} = 1.

- If g ∉ HZY, then u_{H,g,Y} = g^{p|D|}. If H ≰ ZY, since ⟨g⟩Z has order at most p^{3} as g^{p} ∈ Z, we have

|D| = |P : H⟨g⟩ZY | ⩾ p^{2l+2}/p·p^{3}·p^{l} = p^{l−2},

If we now assume H ⩽ Y, then

|D| = |C_{P} (H) : ⟨g⟩ZY | ⩾ p^{2l+1}/p^{3}·p^{l} = p^{l−2}

again. Hence then |D| is a multiple of p if l ⩾ 3, and u_{H,g,Y} = g^{p|D|} = 1.

So u_{H,g,Y} = 1 if P is almost extra-special, unless possibly if P has order p^{6}.

So we are left with the special cases where P is either extra-special of order p^{5} and exponent p^{2}, or almost extra-special of order p^{6}. In both cases, we will use the next lemma, where the notation is again as in 3.15.

3.21. Lemma. Let P be extra-special of order p^{5} and exponent p^{2}, or almost extraspecial of order p^{6}. Then there exist a ∈ X_{0} and b ∈ Y with the following properties:

1. The elements a and b both have order p, and centralize C.

2. The group U = ⟨a, b⟩ is extra-special of order p^{3} and exponent p. Its center is equal to N.

3. For α ∈ {0, …, p − 1}, set H_{α} = ⟨ab^{α}⟩. Also set H_{∞} = ⟨b⟩, and denote by 𝕃 the set {0, …, p−1, ∞}. Then ⟨H_{α}, H_{β}⟩ = U for any pair (α, β) of distinct elements of 𝕃, and the map α ∈ 𝕃 ↦ H_{α}N is a bijection from 𝕃 to the set of subgroups of order p^{2} of U.

Proof. In both cases Y ≅ (C_{p})^{2}. Since [C, Y] ⩽ Z ⩽ C, the group Y normalizes C, and C_{Y} (C) ≠ 1 since the automorphism group of C is cyclic. We choose a non trivial element b of C_{Y} (C). Then b does not centralize X_{0}, for otherwise b centralizes CX_{0} = X, contradicting C_{P} (X) = X. Hence we can choose a ∈ X_{0} such that [a, b] ≠ 1. Then the group U = ⟨a, b⟩ is extra-special of order p^{3} and exponent p, as a and b have order p and [a, b] generates N. In particular N is equal to the center of U.

The groups H_{α}, for α ∈ 𝕃, all have order p, and are different from N. So the groups H_{α}N are subgroups of order p^{2} of U. Clearly ⟨H_{α}, H_{β}⟩ = U for any distinct element α, β of 𝕃, hence the subgroups H_{α}N, for α ∈ 𝕃, are all distinct. This completes the proof, as U has exactly p + 1 = |𝕃| subgroups of order p^{2}.

We now proceed with the proof of the above mentioned two special cases.

3.22. Proposition.

1. Let P be extra-special of order p^{5} and exponent p^{2}. Then the group K of Theorem 3.16 is trivial.

2. Let P be almost extra-special of order at least p^{6}. Then the group K of Theorem 3.16 has order p.

Proof. We keep the notation of Lemma 3.21.

1) Suppose first that P is extra-special of order p^{5} and exponent p^{2}. Then we have X = C × X_{0}, where C ≅ C_{p^{2}} contains the center Z = N of order p of P, and X_{0} ≅ C_{p}. On the other hand Y ≅ (C_{p})^{2} in this case. We will build an element w in the kernel of e : R → \overline{R} with non trivial component w_{Y} ∈ Z. Then w_{Y} will generate Z, so K will be trivial in this case, as stated.

In order to build w, we first fix a generator g of C. Next we consider the set 𝕃 and the subgroups H_{α} of C_{P} (C) = C_{P} (g), for α ∈ 𝕃, introduced in Lemma 3.21. We now set

w = (u_{1,g})^{−1} ∏_{α∈𝕃} u_{H_{α},g}.

For an arbitrary cyclic subgroup H of C_{P} (g), the component u_{H,g,S} of u_{H,g} at a genetic subgroup S of P is given by Formula 2.5. When |P : S| = p, this formula comes down to

u_{H,g,S} = \{ \begin{matrix}1 & if H ≰ S, \\ \overline{g} & otherwise, \end{matrix}

where \overline{g} is the image of g in P/S. It follows that the component w_{S} of w at S is equal to (\overline{g})^{−1+n_{S}}, where n_{S} = |\{α ∈ 𝕃 | H_{α} ⩽ S\}|. Then either S ⩾ U = ⟨a, b⟩, and S ⩾ H_{α} for all α ∈ 𝕃, so n_{S} = p + 1, and w_{S} = (\overline{g})^{p} = 1, since P/S has order p; or S ≱ U, and then S ∩ U is a subgroup of index p of U. So |S ∩ U| = p^{2}, and S ∩ U = H_{α}Z for a unique α ∈ 𝕃. Hence H_{α} ⩽ S, and H_{β} ≰ S for β ∈ 𝕃 − {α}. It follows that n_{S} = 1 in this case, so w_{S} = (\overline{g})^{−1+1} = 1 again. This shows that w is in the kernel of the map e : R → \overline{R}.

We now compute the component w_{Y} of w indexed by Y, using 3.20: we consider a cyclic subgroup H of order at most p of C_{P} (g). Then g ∉ HZY, because HZY has exponent p. Then u_{H,g,Y} = g^{p|D|}, where D is a set of representatives of those double cosets H⟨g⟩tZY such that H^{t} ∩ ZY ⩽ Y.

We know moreover that |D| = |P : H⟨g⟩ZY | = |P : HCY | if H ≰ ZY, since Z ⩽ C = ⟨g⟩, and that |D| = |C_{P} (H) : ⟨g⟩ZY | = |C_{P} (H) : CY | if H ⩽ Y.

If H = 1, then |D| = |P |/(|C||Y |) = p^{5}/(p^{2}·p^{2}) = p, so u_{H,g,Y} = g^{p^{2}} = 1. If H = H_{α}, for α ∈ 𝕃 − {∞}, then |D| = |P : H_{α}CY | = p^{5}/(p·p^{2}·p^{2}) = 1, for H_{α} ≰ CY. Then u_{H_{α},g,Y} = g^{p}. Finally, if H = H_{∞}, we have |D| = |C_{P} (H_{∞}) : CY | = p^{4}/(p^{2}·p^{2}) = 1, so u_{H_{α},g,Y} = g^{p} again.

This gives w_{Y} = 1^{−1} ∏_{α∈𝕃} g^{p} = g^{p(p+1)} = g^{p}, and g^{p} is a generator of Z. So we have

indeed built an element w in the kernel of e : R → \overline{R} such that the component w_{Y} generates Z. It follows that the group K is trivial in this case, as claimed.

2) Suppose now that P is almost extra-special of order p^{6}. Then we have C = Z ≅ C_{p^{2}} > P′ = N. In this case, let g ∈ P and H be a subgroup of order 1 or p of C_{P} (H), Equation 3.20 shows that u_{H,g,Y} is equal to (v_{H,g})^{|D|} for some v_{H,g} ∈ Z, where D is a set of representatives of double cosets H⟨g⟩tZY such that H^{t} ∩ ZY ⩽ Y. We claim that u_{H,g,Y} ∈ N. Indeed, if g ∉ HZY, then v_{H,g} = g^{p} ∈ N. So we can assume that g ∈ HZY. Then if H ≰ ZY, we have |D| = |P : H⟨g⟩ZY | = |P : HZY | ⩾ p^{6}/(p·p^{2}·p^{2}) = p. If H ⩽ Y, then g ∈ HZY = ZY, and |D| = |C_{P} (H) : ZY | ⩾ p^{5}/(p^{2}·p^{2}) = p. In both cases |D| is a multiple of p, so u_{H,g,Y} = (v_{H,g})^{|D|} ∈ N again, as claimed.

As in the proof of Assertion 1, we will build an element w in the kernel of e : R → \overline{R} such that w_{Y} generates N. This will show that K ≅ Z/N ≅ C_{p}, as claimed. We first choose a generator g of Z = C. The formal definition of w is then the same as in the proof of Assertion 1:

w = (u_{1,g})^{−1} ∏_{α∈𝕃} u_{H_{α},g}.

The computation of the component w_{S} for a subgroup S of index p of P is also very similar to what we did for Assertion 1: for a subgroup S of index p of P, we have w_{S} = (\overline{g})^{−1+n_{S}}, where n_{S} = |\{α ∈ 𝕃 | H_{α} ⩽ S\}|. We have n_{S} = 1 or n_{S} = p + 1, thus w_{S} = 1, and w lies in the kernel of e : R → \overline{R}.

As for the component w_{Y} of w, we observe that u_{H,g,Y} = g^{|D|} since g ∈ Z, where D is as above. If H = 1, then |D| = |P : ZY | = p^{2}, and u_{1,g,Y} = g^{|P:ZY |} = g^{p^{2}} = 1. For α ∈ 𝕃 − {∞}, we have |D| = |P : H_{α}⟨g⟩ZY | = |P/H_{α}ZY | = p^{6}/p·p^{2}·p^{2} = p, so u_{H_{α},g,Y} = g^{p}. Finally, for H = H_{∞} ⩽ Y, we have |D| = |C_{P} (H_{∞}) : ⟨g⟩ZY | = |C_{P} (H_{∞}) : ZY | = p^{5}/(p^{2}·p^{2}) = p, so u_{H_{∞},g,Y} = g^{p} again.

It follows that w_{Y} = 1^{−1}(g^{p})^{p+1} = g^{p^{2}+p} = g^{p}, so w_{Y} generates N, as was to be shown.

References

[1] R. C. Alperin, R. K. Dennis, R. Oliver, and M. R. Stein. SK_{1} of finite abelian groups, II. Invent. Math., 87:253–302, 1987.

[2] L. Barker. Genotypes of irreducible representations of finite p-groups. Journal of Algebra, 306:655–681, 2007.

[3] D. J. Benson. Representations and cohomology II, volume 31 of Cambridge studies in advanced mathematics. Cambridge University Press, 1991.

[4] S. Bouc. K-theory, genotypes, and p-biset functors. In preparation.

[5] S. Bouc. The Dade group of a p-group. Inv. Math., 164:189–231, 2006.

[6] S. Bouc. The functor of units of Burnside rings for p-groups. Comm. Math. Helv., 82:583–615, 2007.

[7] S. Bouc. Biset functors for finite groups. Springer, Berlin, 2010.

[8] S. Bouc. Fast decomposition of p-groups in the Roquette category, for p > 2. In RIMS Kôkyûroku, volume 1872, pages 113–121, 2014. arXiv:1403.6092.

[9] S. Bouc and N. Mazza. The Dade group of (almost) extraspecial p-groups. Journal of Pure and Applied Algebra, 192:21–51, 2004.

[10] J. F. Carlson and J. Thévenaz. Torsion endo-trivial modules. Algebras and Representation Theory, 3:303–335, 2000.

[11] K. Doerk and T. Hawkes. Finite soluble groups. Walter de Gruyter, Berlin, 1992.

[12] B. Huppert, Endliche Gruppen I, Springer-Verlag Berlin Heidelberg New York, Die Grundlehren der mathematischen Wissenschaften, vol. 134, 1967.

[13] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.8.5,

    2016. (http://www.gap-system.org).

[14] D. Gorenstein. Finite groups. Chelsea Publishing, New York, 1968.

[15] J. Guaschi, D. Juan-Pineda, and S. Millán López. The lower algebraic K-theory of the braid groups of the sphere. Preprint, arXiv:1209.4791, 2012.

[16] J.-F. Lafont, B. A. Magurn, and I. J. Ortiz. Lower algebraic K-theory of certain reflection groups. Proceedings of the Cambridge Philosophical Society, 148:193– 226, 2010.

[17] R. Oliver. Whitehead groups of finite groups. Cambridge University Press, U.K., 1988.

[18] N. Romero. Computing Whitehead groups using genetic bases. Journal of algebra, 450:646–666, 2016.

[19] P. Roquette. Realisierung von Darstellungen endlicher nilpotenter Gruppen. Arch. Math., 9:241–250, 1958.

[20] J.-P. Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977.

[21] F. Ushitaki. SK_{1}(ℤ[G]) of finite solvable groups which act linearly and freely on spheres. Osaka Journal of Mathematics, 28(1):117–127, 1991.

[22] F. Ushitaki. A generalization of a theorem of Milnor. Osaka Journal of Mathematics, 31:430–415, 1994.

[23] J. H. C. Whitehead. Simple homotopy types. American Journal of Mathematics, 72:1–57, 1950.

Serge Bouc, CNRS-LAMFA, 33 rue St Leu, 80039, Amiens, France.

serge.bouc@u-picardie.fr

Nadia Romero, DEMAT, UGTO, Jalisco s/n, Mineral de Valenciana, 36240, Guanajuato, Gto., Mexico.

nadia.romero@ugto.mx