The functor of rational representations for p-groups

Serge Bouc

Institut de Mathématiques de Jussieu Université Paris 7-Denis Diderot, 75251, Paris Cedex 05, France email: bouc@math.jussieu.fr

Abstract : Let k be a field of characteristic p > 0. Let C_{p,k} be the category whose objects are the finite p-groups, morphisms are the k-linear combinations of bisets, and composition of morphisms is obtained by k-linear extension from the usual product of bisets. Let F_{p,k} denote the category of k-linear functors from C_{p,k} to the category of k-vector spaces.

This paper investigates the structure of the functor kR_{ℚ} mapping a pgroup P to kR_{ℚ}(P) = k ⊗_{ℤ} R_{ℚ}(P), as an object of F_{p,k}, where R_{ℚ}(P) is the Grothendieck group of the category of finite dimensional ℚP -modules. The main result is an explicit description of the lattice of all subfunctors of kR_{ℚ}. In particular for p odd, it is shown that kR_{ℚ} is a uniserial object in F_{p,k}. For p = 2, the lattice of subfunctors of kR_{ℚ} can be described as the lattice of closed subsets of a graph whose vertices are the 2-groups of normal 2-rank 1.

In both cases a composition series of kR_{ℚ} is obtained, which leads to a formula giving the dimension of the evaluations S_{Q,k}(P) of the simple objects S_{Q,k} of F_{p,k} associated to p-groups Q of normal p-rank 1, different from C_{p}. This formula can be phrased in terms of rational representations of P, but also in terms of the geometry of the lattice of subgroups of P, using the notions of basic subgroups and origins. For example, if p = 2, the dimension of S_{1,k}(P) is equal to the number of absolutely irreducible ℚP -modules.

AMS classification : 20C15, 19A22

Keywords : rational representation p-group biset functor 2

The functor of rational representations for p-groups

Serge Bouc

Institut de Mathématiques de Jussieu Université Paris 7-Denis Diderot, 75251, Paris Cedex 05, France email: bouc@math.jussieu.fr

1 Introduction

The formalism of bisets for finite groups, exposed in [1], provides a single framework involving the usual operations of induction, restriction, inflation, and deflation naturally associated to the usual representation groups of finite groups. With this formalism, these representation groups appear as functors from a suitable category, whose objects are (some specific classes of) finite groups, and morphisms are (some classes of) bisets, or linear combination of bisets, with values in the category of abelian groups, or more generally of modules over some fixed commutative ring.

The context of this article is the following special case, also exposed in [4] (Section 7) : let p be a prime number, and k be a field. Let C = C_{p,k} denote the following category :

• The objects of C are the finite p-groups.

• If P and Q are finite p-groups, then

Hom_{C}(P, Q) = k ⊗_{ℤ} B(Q × P^{op}) ,

where B(Q × P^{op}) is the Burnside group of Q × P^{op}, viewed as the Grothendieck group of the category of finite Q-sets-P.

• The composition of morphisms in C is obtained by k-bilinearity from the usual product of bisets : if P, Q and R are finite p-groups, if X is a Q-set-P, and Y is an R-set-Q, then the composition of X ∈ Hom_{C}(P, Q) and Y ∈ Hom_{C}(Q, R) is equal to Y ∘ X = Y ×_{Q} X.

Let F = F_{p,k} denote the category of k-linear functors from C_{p,k} to the category of k-vector spaces. Then F is an abelian category. The simple objects of F are parametrized by pairs (Q, V) consisting of a finite p-group Q and a simple kOut(Q)-module V ([1] Proposition 2 page 678). The simple functor corresponding to the pair (Q, V) is denoted by S_{Q,V}.

Date : October 9, 2003

Most of the natural representation groups, such as the groups R_{ℚ}(P) of rational representations of the p-group P, or the Burnside group B(P), provide examples of objects of F, after tensoring with k : in particular, the correspondence P ↦ kR_{ℚ}(P) = k ⊗_{ℤ} R_{ℚ}(P) is a k-linear functor on C, with values in k-vector spaces. If Q and P are finite p-groups, and if U is a finite Q-set-P, viewed as an element of Hom_{C}(P, Q), then

kR_{ℚ}(U) : kR_{ℚ}(P) → kR_{ℚ}(Q)

is the map obtained by k-linearity from the map sending the class of the ℚP -module V to the class of the ℚQ-module ℚU ⊗_{ℚP} V.

Remark 1.1 The reason for restricting the attention to p-groups is twofold : firstly, for p-groups, the Ritter-Segal theorem holds : any rational representation of a p-group is a (virtual) permutation representation, thus combinatorial or geometric properties of p-groups give information on their rational representations. Secondly, some important functors appearing naturally in modular representations of finite groups are defined only for p-groups : the main examples here are provided by groups of endo-permutation modules (see [4] for details), i.e. subgroups of the Dade group, such as the group of relative syzygies ([2]), which are functorial in the above sense.

Example 1.2 Recall in particular, if Q is a subgroup of P, then the operation associated to the set U = P, viewed as a Q-set-P, is ordinary restriction Res_{Q}^{P} of modules. If U is viewed as a P -set-Q, the corresponding operation is induction Ind_{Q}^{P} of modules. Similarly, if Q = P/N is a factor group of P, then the set U = Q, viewed as a P -set-Q, corresponds to inflation Inf_{Q}^{P} of modules. And when U is viewed as a Q-set-P, the associated operation is called deflation, and denoted by Def_{Q}^{P}. The effect on a module is the construction of co-invariants by N. Finally, if φ : P → Q is a group isomorphism, then the obvious associated operation of change of group is denoted by Iso_{P}^{Q}, and corresponds to the set U = Q, viewed as a Q-set-P.

The structure of kR_{ℚ} as an object of F_{p,k} depends on the characteristic q of k. The case q ≠ p is solved by Theorem 8.2 and Corollaries 8.3 and 8.4 of [4] : in this case, if q does not divide p − 1, the functor kR_{ℚ} is simple, isomorphic to S_{1,k}. And if q divides p − 1, then kR_{ℚ} has a unique non zero proper subfunctor F, isomorphic to S_{C_{p},k}, and the quotient kR_{ℚ}/F is isomorphic to S_{1,k}.

This article deals with the remaining case q = p. The situation in this case is quite different. The possible methods to attack this question are also completely different : the main tool in [4] was the fact that the Burnside algebra kB(P) = k ⊗_{ℤ} B(P) of a p-group P is semi-simple if q ≠ p, with explicit formulae for primitive idempotents. In the case q = p on the contrary, the algebra kB(P) is local, and it has no non-trivial idempotents.

The main results stated here (Theorem 6.1, Theorem 6.2, Corollary 6.5) describe the lattice of subfunctors of kR_{ℚ}, in terms of groups of normal p-rank 1. Any such group P has a unique irreducible faithful rational module Φ_{P}, whose image in kR_{ℚ}(P) generates a subfunctor H_{P} of kR_{ℚ}. Then any subfunctor of kR_{ℚ} is the sum of those functors H_{P} it contains. When p is odd, the situation is simple, since kR_{ℚ} is uniserial. When p = 2, the lattice of subfunctors of kR_{ℚ} can be described as the lattice of closed subsets of an explicit graph, whose vertices are the isomorphism classes of groups of normal p-rank 1.

The main consequence of this description is the following :

Theorem 1.3 Let k be a field of characteristic p > 0.

1. If p ≠ 2, then the functor kR_{ℚ} is a uniserial object of F_{p,k}. If F_{0} = kR_{ℚ} ⊃ F_{1} ⊃ F_{2} ⊃ … ⊃ F_{n} ⊃ … is the set of its non-zero subfunctors, then

F_{0}/F_{1} ≅ S_{1,k} F_{i}/F_{i+1} ≅ S_{C_{p^{i+1}},k} for i ≥ 1 ,

where C_{p^{i+1}} denotes a cyclic group of order p^{i+1}.

2. If p = 2, then the functor kR_{ℚ} has a filtration F_{0} = kR_{ℚ} ⊃ F_{1} ⊃ F_{2} ⊃ … ⊃ F_{n} ⊃ … by subfunctors F_{n} such that ∩_{n≥0} F_{n} = 0 and

F_{0}/F_{1} ≅ S_{1,k}
F_{1}/F_{2} ≅ S_{C_{4},k} ⊕ S_{D_{16},k} ⊕ S_{SD_{16},k}
F_{i}/F_{i+1} ≅ S_{C_{2^{i+1}},k} ⊕ S_{Q_{2^{i+1}},k} ⊕ S_{D_{2^{i+3}},k} ⊕ S_{SD_{2^{i+3}},k} if i ≥ 2,

where C_{2^{i}}, Q_{2^{i}}, D_{2^{i}} and SD_{2^{i}} denote respectively the cyclic, generalized quaternion, dihedral, and semi-dihedral group of order 2^{i}.

This theorem can be considered as a functorial refinement of methods initiating in P. Roquette’s work ([7]). Assertion 1 has been conjectured by I. Bourizk ([5] Conjecture 1).

For p = 2, the description of the lattice of subfunctors of kR_{ℚ} shows in particular that the subfunctor H_{Q_{8}} of kR_{ℚ} attached to the quaternion group Q_{8} is a uniserial object of F. This was conjecture 2 in [5].

The other main result of this article (Theorem 5.12) is a computation of the k-dimension of the evaluation of the simple functors S_{P,k}, when P is a p-group of normal p-rank 1, different from C_{p} (the notion of type of an irreducible rational module for a p-group is introduced in Definition 3.5) :

Theorem 1.4 Let Q be a finite p-group.

1. The dimension of S_{1,k}(Q) is equal to

dim_{k} S_{1,k}(Q) = |{V ∈ Irr_{ℚ}(Q) | V has type 1 or C_{p}}|
= |{V ∈ Irr_{ℚ}(Q) | ⟨V, V ⟩_{Q} ≤ p − 1}|
= |{V ∈ Irr_{ℚ}(Q) | ⟨V, V ⟩_{Q} ≢ 0 (p)}|

2. If P has normal p-rank 1, and if |P | ≥ p^{2}, then

dim_{k} S_{P,k}(Q) = |{V ∈ Irr_{ℚ}(Q) | V has type P }| .

This gives a partial answer to a question already raised in [1] (Section 8, page 713).

Remark 1.5 The particular status of C_{p} in this theorem will be explained in Remark 5.7.

The paper is organized as follows : in section 2, I will recall some results on the rational irreducible representations of p-groups. Section 3 deals with the irreducible representations of p-groups of normal p-rank 1.

This leads to the definition of the subfunctors H_{P} of kR_{ℚ}, in section 4. In section 5, it is shown that the functor H_{P} has a unique simple quotient, isomorphic to S_{P,k} if |P | ≠ p. An expression of the k-dimension of S_{P,k}(Q) is given, for a p-group Q and a p-group P of normal p-rank 1 (different from C_{p}), in terms of rational representations of Q.

Section 6 describes the possible mutual inclusions of subfunctors H_{P}, and this gives the structure of the lattice of subfunctors of kR_{ℚ}. The last section exposes an interpretation of the dimension of the evaluations of the simple functors obtained in section 4, in terms of the bilinear forms introduced in [1] Proposition 16 page 717. The result is an expression of these dimensions purely in terms of the structure of the lattice of subgroups of the given pgroup, without any reference to rational representations.

2 Rational representations of p-groups

Recall the following theorem from [3] :

Theorem 2.1 ([3] Theorem 1) Let p be a prime number, and P be a finite p-group. If V is a non-trivial simple ℚP -module, then there exist subgroups R ⊃ Q of P, with |R : Q| = p, and an isomorphism of ℚP -modules

V ≅ Ind_{R}^{P}Inf_{R/Q}^{R}Ω_{R/Q}

where Ω_{R/Q} is the augmentation ideal of the group algebra ℚ(R/Q). In other words there is an exact sequence of ℚP -modules

0 → V → ℚ(P/Q) → ℚ(P/R) → 0

where the map ℚ(P/Q) → ℚ(P/R) is the natural projection. In particular in R_{ℚ}(P)

V = ℚ(P/Q) − ℚ(P/R)

Conversely, the following proposition is a test for irreducibility of the module Ind_{R}^{P}Inf_{R/Q}^{R}Ω_{R/Q} :

Proposition 2.2 ([3] Proposition 4) Let p be a prime number, and P be a p-group. Let R ⊃ Q be subgroups of P, with |R : Q| = p. Then the following conditions are equivalent:

1. The module Ind_{R}^{P}Inf_{R/Q}^{R}Ω_{R/Q} is an irreducible ℚP -module.

2. If S is any subgroup of P such that R ∩ S ⊆ Q, then |S| < |R|.

3. The group N_{P} (Q)/Q is cyclic or generalized quaternion, the group R/Q is its unique subgroup of order p, and if S is any subgroup of P such that |S| > |Q|, then S ∩ N_{P} (Q) ⊈ Q.

This proposition shows in particular that R is determined by Q.

Definition 2.3 Let P be a finite p-group. By definition, a basic subgroup of P is a subgroup Q of P such that the following two conditions hold :

1. The group N_{P} (Q)/Q is cyclic or generalized quaternion.

2. If S is any subgroup of P such that |S| > |Q|, then S ∩ N_{P} (Q) ⊈ Q.

The group P itself is a basic subgroup of P.

Notation 2.4 Let P be a finite p-group.

• Let Irr_{ℚ}(P) denote the set of isomorphism classes of irreducible ℚP - modules, and let B_{P} denote the set of basic subgroups of P.

• If Q is a proper basic subgroup of P, then let V_{Q} denote the simple ℚP - module IndP ; Q̃InfQ̃ ; Q̃/QΩ_{Q̃/Q}, where Q̃/Q is the unique subgroup of order p of N_{P} (Q)/Q. Also let V_{P} denote the trivial ℚP -module. The simple module V_{Q} is called the module associated to the basic subgroup Q.

• If V and W are finite dimensional ℚP -modules, let

⟨V, W ⟩_{P} = dim_{ℚ} Hom_{ℚP}(V, W) .

This extends uniquely to a bilinear form on R_{ℚ}(P), with values in ℤ, still denoted by ⟨ , ⟩_{P}.

• If V is a simple ℚP -module, denote by m(V, W) the multiplicity of V in W, i.e. the number of times V appears in a decomposition of W as a direct sum of simple ℚP -modules.

• If S and Q are subgroups of P, set

I_{P} (S, Q) = {x ∈ P | S^{x} ∩ N_{P} (Q) ⊆ Q}

and

I_{P} (S, Q) = S\I_{P} (S, Q)/N_{P} (Q) .

The following proposition lists some properties of basic subgroups and associated representations :

Proposition 2.5 Let P be a finite p-group.

1. If Q is a proper basic subgroup of P, and S is any subgroup of P, then

I_{P} (S, Q) = {x ∈ P | S^{x} ∩ Q̃ ⊆ Q} ,

and

⟨ℚ(P/S), V_{Q}⟩_{P} =(p − 1)|{x ∈ S\P/Q̃ | S^{x} ∩ Q̃ ⊆ Q}|
=(p − 1)\frac{|N_{P} (Q)/Q|}{p}|I_{P} (S, Q)|

2. If Q is a basic subgroup of P, and if S is any subgroup of P, then

m(V_{Q}, ℚ(P/S)) = \frac{|I_{P} (S, Q)|}{|I_{P} (Q, Q)|}

3. If Q is a basic subgroup of P, then the cardinality of I_{P} (Q, Q) is a power of p. If moreover Q ≠ P, then

⟨V_{Q}, V_{Q}⟩_{P} = (p − 1)\frac{|N_{P} (Q)/Q|}{p}|I_{P} (Q, Q)| .

4. If Q and Q′ are basic subgroups of P, then the corresponding simple ℚP -modules V_{Q} and V_{Q′} are isomorphic if and only if |Q| = |Q′| and I_{P} (Q, Q′) ≠ ∅.

Proof : Observe first that if Q is a proper basic subgroup of P, if x ∈ P and S^{x} ∩ N_{P} (Q) ⊆ Q, then S^{x} ∩ Q̃ ⊆ Q, since Q̃ ⊆ N_{P} (Q). Conversely, if S^{x} ∩ N_{P} (Q) ⊈ Q, then Q̃ ⊆ (S^{x} ∩ N_{P} (Q))Q, since Q̃/Q is the only subgroup of order p of N_{P} (Q)/Q. Thus

Q̃ = ((S^{x} ∩ N_{P} (Q)) ∩ Q̃)Q = (S^{x} ∩ Q̃).Q

and S^{x} ∩ Q̃ ⊈ Q. This shows that

S^{x} ∩ Q̃ ⊆ Q if and only if S^{x} ∩ N_{P} (Q) ⊆ Q .

Now recall that if R and S are subgroups of P, then

⟨ℚ(P/R), ℚ(P/S)⟩_{P} = |R\P/Q| .

If Q is a proper basic subgroup of P, since V_{Q} = ℚ(P/Q) − ℚ(P/Q̃) in R_{ℚ}(P), it follows that

⟨ℚ(P/S), V_{Q}⟩_{P} = |S\P/Q| − |S\P/Q̃| .

Let x ∈ P. There are two cases : either S^{x}∩Q̃ ⊈ Q, and then Q̃ = (S^{x}∩Q̃)Q since |Q̃ : Q| = p. Hence Q̃ ⊆ S^{x}.Q in this case, and SxQ̃ = SxQ. Or S^{x} ∩ Q̃ ⊆ Q, and in this case

|SxQ̃| = \frac{|S||Q̃|}{|S^{x} ∩ Q̃|} = p\frac{|S||Q|}{|S^{x} ∩ Q|} = p|SxQ| .

This shows that the map φ : S\P/Q → S\P/Q̃ sending SxQ to SxQ̃ is such that |φ^{−1}(u)| = 1 or p, for all u ∈ S\P/Q̃. Hence

|S\P/Q| − |S\P/Q̃| = ∑_{u∈S\P/Q̃} (|φ^{−1}(u)| − 1)
= (p − 1)|{u ∈ S\P/Q̃ | |φ^{−1}(u)| = p}|

and finally

⟨ℚ(P/S), V_{Q}⟩_{P} = (p − 1)|{x ∈ S\P/Q̃ | S^{x} ∩ Q̃ ⊆ Q}|  (2.6)

proving the second formula in Assertion 1. Now the group N_{P} (Q) normalizes Q̃, and acts on the right on the set

U = {x ∈ S\P/Q̃ | S^{x} ∩ Q̃ ⊆ Q}

by right multiplication. The stabilizer in N_{P} (Q) of SxQ̃ is the set of elements n ∈ N_{P} (Q) such that SxnQ̃ = SxQ̃, or equivalently

n ∈ S^{x}.Q̃ ∩ N_{P} (Q) = (S^{x} ∩ N_{P} (Q)).Q̃ = Q̃ .

In other words, the group N_{P} (Q)/Q̃ acts freely on U, and the set of its orbits is precisely the set of elements x ∈ S\P/N_{P} (Q) such that S^{x} ∩ N_{P} (Q) ⊆ Q. Hence

|U| = |N_{P} (Q)/Q̃||I_{P} (S, Q)|

proving the third formula of Assertion 1.

It follows in particular that ⟨ℚ(P/S), V_{Q}⟩_{P} = 0 if |S| > |Q|, by definition of a basic subgroup. Thus

⟨V_{Q}, V_{Q}⟩_{P} = ⟨ℚ(P/Q), V_{Q}⟩_{P} − ⟨ℚ(P/Q̃), V_{Q}⟩_{P} = ⟨ℚ(P/Q), V_{Q}⟩_{P}
= (p − 1)\frac{|N_{P} (Q)/Q|}{p}|I_{P} (Q, Q)|

proving the second part of Assertion 3.

Now for any ℚP -module W

⟨W, V_{Q}⟩_{P} = m(V_{Q}, W)⟨V_{Q}, V_{Q}⟩_{P}

since the irreducible modules are mutually orthogonal for ⟨ , ⟩_{P}. Taking W = ℚ(P/S) gives Assertion 2, in the case Q ≠ P. The case Q = P is obvious, since the trivial module V_{P} has multiplicity 1 in ℚ(P/S).

Taking now S = 1 in Assertion 2 gives

m(V_{Q}, ℚ(P/1)) = \frac{|P : N_{P} (Q)|}{|I_{P} (Q, Q)|}

showing that the denominator is a power of p, and completing the proof of Assertion 3 in the case Q ≠ P. The case Q = P is trivial.

Finally, suppose that Q and Q′ are basic subgroups of P such that the ℚP - modules V_{Q} and V_{Q′} are isomorphic. Then Q = P if and only if Q′ = P, since V_{Q} is not the trivial module for Q ≠ P. And if Q and Q′ are proper basic subgroups of P, then |Q| = |Q′|, since dim_{ℚ} V_{Q} = |P : Q|(p − 1)/p = dim_{ℚ} V_{Q′} = |P : Q′|(p − 1)/p. Hence |Q̃| > |Q′|, and ⟨V_{Q}, V_{Q′}⟩_{P} = ⟨ℚ(P/Q), V_{Q′}⟩_{P}. This is non zero if V_{Q} ≅ V_{Q′}. By Assertion 1, the set |I_{P} (Q, Q′)| is not empty.

Conversely, if |Q| = |Q′|, then either Q = Q′ = P, and V_{Q} = V_{Q′} = V_{P}, or Q and Q′ are proper basic subgroups of P. Since |Q̃| > |Q′|, it follows as above that ⟨V_{Q}, V_{Q′}⟩_{P} = ⟨ℚ(P/Q), V_{Q′}⟩_{P}, and this is non zero by Assertion 1, if moreover |I_{P} (Q, Q′)| ≠ ∅. Hence V_{Q} ≅ V_{Q′} since V_{Q} and V_{Q′} are both simple. This completes the proof of Assertion 4.

Notation 2.7 If P is a finite p-group, define the following relation on B_{P} :

Q ≑_{P} Q′ ⟺^{def.} |Q| = |Q′| and I_{P} (Q, Q′) ≠ ∅ .

Theorem 2.8 Let P be a finite p-group.

1. The relation ≑_{P} is an equivalence relation on B_{P}.

2. The correspondence Q ∈ B_{P} ↦ V_{Q} ∈ Irr_{ℚ}(P) induces a one to one correspondence between the set of equivalence classes of basic subgroups of P for the relation ≑_{P} and the set of isomorphism classes of rational irreducible representations of P.

Proof : Assertion 1 follows from Assertion 4 of Proposition 2.5. The same argument shows that the correspondence Q ∈ B_{P} ↦ V_{Q} ∈ Irr_{ℚ}(P) induces an injective correspondence between the set of equivalence classes of basic subgroups of P for the relation ≑ and the set Irr_{ℚ}(P). This induced correspondence is moreover surjective, since any irreducible ℚP -module is isomorphic to V_{Q}, for some basic subgroup Q of P, by Theorem 2.1 and Notation 2.4.

Remark 2.9 It it easy to check directly (i.e. without using the modules V_{Q}) that the relation ≑_{P} is reflexive and symmetric, but it’s far from obvious to prove transitivity without this.

3 Groups of normal p-rank 1

Definition 3.1 A p-group P is said to have normal p-rank 1 if it does not have any normal subgroup isomorphic to (C_{p})^{2}. Denote by N the class of finite p-groups of normal p-rank 1, and by [N] a set of isomorphism classes of elements of N. More generally, if A is a subclass of N, set [A] = A∩[N].

If P ∈ N, then by Theorem 4.10 of Chapter 5 of [6], the group P is cyclic if p is odd, and if p = 2, the group P is cyclic, or generalized quaternion of order at least 8, dihedral of order at least 16, or semi-dihedral of order at least 16.

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}]

This diagram gives a good idea of the general case, quoted without proof in the following lemma :

Lemma 3.2 Let P a non-cyclic group in N. Then p = 2, and P has exactly 3 maximal subgroups A, B, and C. The group C is cyclic, and moreover :

1. If P is dihedral, then A and B are dihedral.

2. If P ≅ Q_{8}, then A and B are cyclic.

3. If P is generalized quaternion of order at least 16, then A and B are generalized quaternion.

4. If P is semi-dihedral, then A is dihedral and B is generalized quaternion.

There are 0,1, or 2 conjugacy classes of non-central involutions in P, according to P being generalized quaternion, semi-dihedral, or dihedral.

Remark 3.3 It follows in particular that if R is a subgroup of P ∈ N, then R ∈ N, except if p = 2 and R ≅ (C_{2})^{2} or R ≅ D_{8}.

The following theorem gives another way, different from Theorem 2.8, to build irreducible representations of a finite p-group. It is more or less well known, and already implicit in Roquette’s methods ([7]) :

Theorem 3.4 Let p be a prime number, and P be a p-group. If V is an irreducible ℚP -module, then there exist subgroups S and T of P, with S \underline{◁} T, and a faithful irreducible ℚ(T/S)-module W such that :

1. The module V is isomorphic to Ind_{T}^{P} Inf_{T/S}^{T}W.

2. This isomorphism induces an isomorphism of ℚ-algebras

End_{ℚP}(V) ≅ End_{ℚ(T/S)}(W)

3. The group T/S has normal p-rank 1.

Proof : For convenience, let me sketch a proof, which is very similar to the proof of Theorem 2.1 (see [3]), and goes by induction on the order of P. One can suppose V faithful, since otherwise V is inflated from a factor group of P, for which the theorem holds by induction. Moreover inflation preserves the required properties.

Now if P has an elementary abelian normal subgroup E ≅ (C_{p})^{2}, let L be a direct summand of Res_{E}^{P}V. Let I denote the inertial subgroup of L in P, and let L̃ denote the isotypic component of L in Res_{E}^{P}V. Then V ≅ Ind_{I}^{P} L̃ by Clifford theory. Moreover I contains the centralizer C_{P} (E) of E, which has index at most p in P, since P/C_{P} (E) is isomorphic to a subgroup of the automorphism group of E. Thus |P : I| ≤ p, and in particular I \underline{◁} P.

If I = E, then L̃ ≅ V is faithful, thus L is faithful. But E has no faithful irreducible rational representations. Hence I has index p in P. Moreover L̃ is an irreducible representation of I, and if x ∈ P is such that ^{x}L̃ = L̃, then ^{x}L ≅ L, thus x ∈ I. It follows that End_{ℚP}(V) ≅ End_{ℚI}(L̃). Now the induction hypothesis applies to I and L̃, completing the proof.

Definition 3.5 In the situation of Theorem 3.4, the pair (T, S) will be called a genetic section of P for V, and if R ≅ T/S is a p-group of normal p-rank 1, then V will be said to have type R.

The set of all sections (T, S) appearing as genetic sections of P for some V ∈ Irr_{ℚ}(P), will be denoted by G_{P}.

Remark 3.6 A natural question is then to ask if the type of an irreducible ℚP -module V is well defined : the answer is affirmative, and a proof will be given in Corollary 5.9. Another more direct proof will be sketched in Remark 3.9, using the following result on the irreducible faithful rational representations of p-groups of normal p-rank 1, which also shows that once the genetic section (T, S) of P for V is given, then the module W appearing in Theorem 3.4 is uniquely determined up to isomorphism :

Proposition 3.7 Let p be a prime number. If P is a finite p-group of normal p-rank 1, then P admits a unique faithful irreducible rational representation, up to isomorphism.

Proof : By Theorem 2.8, this amounts to show that there is a single equivalence class of basic subgroups Q of P (for the relation ≑_{P}) for which V_{Q} is faithful. If P = 1, this is trivial. If not, let Q be a basic subgroup of P. If V_{Q} is faithful, then Q ≠ P. Moreover since V_{Q} = IndP ; Q̃InfQ̃ ; Q̃/QΩ_{Q̃/Q} is faithful, the intersection of conjugates of Q in P is trivial. Equivalently Q intersects the center of P trivially.

If P is cyclic or generalized quaternion, this forces Q = 1. Conversely, in that case, it follows from the definition that the trivial subgroup is basic. The corresponding simple ℚP -module V_{1} is faithful, and it is the only faithful simple ℚP -module in this case, up to isomorphism.

If P is dihedral or semi-dihedral, and if Q intersects the center of P trivially, then Q is trivial, or non-central of order 2. In that case, the trivial subgroup is not basic (since N_{P} (1)/1 = P is neither cyclic nor generalized quaternion !). Hence Q is non central, of order 2. Conversely if Q is noncentral and of order 2, then N_{P} (Q) has order 4, equal to Q.Z, where Z is the center of P. Thus N_{P} (Q)/Q is cyclic, and Q is basic, since any subgroup of P of order at least 4 contains Z.

Now if P is semi-dihedral, then there is a single conjugacy class of noncentral subgroups of order 2 in P. Such a subgroup Q is basic, and V_{Q} is the only faithful simple ℚP -module, up to isomorphism.

And if P is dihedral, there are two conjugacy classes of non-central subgroups of order 2 in P. Let Q and Q′ be such subgroups in different classes. Then N_{P} (Q) has order 4, and is equal to Q.Z, where Z is the center of P. Since Q′ ∩ N_{P} (Q) = 1 ⊆ Q, it follows that Q ≑_{P} Q′. Hence V_{Q} ≅ V_{Q′} is the only faithful simple ℚP -module in that case, up to isomorphism.

Notice that this argument holds also for the dihedral group D_{8} of order 8, showing that it has a unique faithful rational simple module.

Notation 3.8 If R is a finite p-group of normal p-rank 1, or if R ≅ D_{8}, let Φ_{R} denote its unique faithful irreducible rational representation.

If P is a finite p-group, and if (T, S) ∈ G_{P}, let V (T, S) = Ind_{T}^{P} Inf_{T/S}^{T}Φ_{T/S} denote the corresponding simple ℚP -module.

Remark 3.9 One can then prove directly that if P is a p-group, the type of a simple ℚP -module V is well defined, up to isomorphism : indeed, if V has type R, then End_{ℚR}Φ_{R} ≅ End_{ℚP}V. Let (T, S) and (T′, S′) be genetic sections of P for V, and set R = T/S and R′ = T′/S′. Then End_{ℚR}Φ_{R} ≅ End_{ℚR′}Φ_{R′}. But the structure of the algebra End_{ℚR}Φ_{R} (actually a skew-field) can be explicitly obtained from the above description of Φ_{R}, and then it is easy to see that for R and R′ in N, the algebra isomorphism End_{ℚR}Φ_{R} ≅ End_{ℚR′}Φ_{R′} is equivalent to the group isomorphism R ≅ R′.

There is now an obvious equivalence relation on the set G_{P} of genetic sections of a finite p-group P : say that the elements (T, S) and (T′, S′) of G_{P} are equivalent if and only if the corresponding simple ℚP -modules V (T, S) and V (T′, S′) are isomorphic. It turns out that this equivalence relation can be translated in a purely “geometric” form : first recall the following definition from [1] :

Definition 3.10 ([1] page 685) Let P be a p-group. Two pairs (T, S) and (V, U) of subgroups of P, with S \underline{◁} T and U \underline{◁} V are said to be linked (notation (T, S) (V, U)) if

S(T ∩ V) = T  U(T ∩ V) = V  S ∩ V = T ∩ U

or equivalently if T.U = S.V (as subsets of P) and T ∩ U = S ∩ V.

Two such pairs (V, U) and (T, S) are said to be linked modulo P (notation (V, U) _{P} (T, S) if there exists x ∈ P such that (V, U) (^{x}T ,^{x}S).

Theorem 3.11 Let P be a finite p-group.

1. The relation _{P} is an equivalence relation on G_{P}.

2. The correspondence (T, S) ∈ G_{P} ↦ V (T, S) ∈ Irr_{ℚ}(P) induces a one to one correspondence between the set of equivalence classes of genetic sections of P for the relation _{P} and the set of isomorphism classes of irreducible ℚP -modules.

Proof : This is another form of Theorem 7.11, proved in Section 7.

The following shows in particular that if R ∈ N (or if R ≅ D_{8}), then the proper deflations of Φ_{P} are zero :

Lemma 3.12 Let P be a finite group. If N is a non-trivial normal subgroup of P, and Φ is a faithful simple ℚP -module, then

Def_{P/N}^{P}Φ = 0 .

Proof : If V is any ℚ(P/N)-module, then Inf_{P/N}^{P}V is a sum of non-faithful irreducible representations of P. Thus ⟨Φ, Inf_{P/N}^{P}V ⟩_{P} = 0, since Φ is faithful. By adjunction, it follows that ⟨Def_{P/N}^{P}Φ, V ⟩_{P/N} = 0 for any V, hence Def_{P/N}^{P}Φ = 0.

Notation 3.13 Denote by N_{1} the subclass of N consisting of groups of order at least p^{2}.

Lemma 3.14 Let P ∈ N_{1}, and R be a maximal subgroup of P.

1. If P is cyclic or generalized quaternion, then Res_{R}^{P}Φ_{P} = pΦ_{R}.

2. If P is dihedral or semi-dihedral, then

Res_{R}^{P}Φ_{P} = \{ \begin{matrix}Φ_{R} & if R is cyclic or generalized quaternion \\ 2Φ_{R} & if R is dihedral \end{matrix} .

Proof : Let Z denote the only central subgroup of P of order p. There are several cases :

• If P is cyclic or generalized quaternion, then so is R. Moreover Φ_{P} = ℚ(P/1) − ℚ(P/Z). Then clearly

Res_{R}^{P}Φ_{P} = |P : R|(ℚ(R/1) − ℚ(R/Z)) = pΦ_{R}

in this case.

• If p = 2 and P is dihedral or semi-dihedral, then

Φ_{P} = ℚ(P/S) − ℚ(P/SZ) ,

where S is non-central of order 2. If R contains S, then R is dihedral, and

Res_{R}^{P}Φ_{P} = ∑_{x∈P/R} (ℚ(R/^{x}S) − ℚ(R/^{x}SZ)) = 2Φ_{R}

since ^{x}S is a non-central subgroup of R of order 2, and Z is the unique central subgroup of order 2 of R. The conclusion is the same if P is dihedral, and R is dihedral, but does not contain S. In that case indeed, one has that Φ_{P} = ℚ(P/S′) − ℚ(P/S′Z) for a non-central subgroup S′ of P contained in R.

• The only remaining cases are when P is dihedral or semi-dihedral, and R is cyclic or generalized quaternion. In that case

Res_{R}^{P}Φ_{P} = ℚ(R/1) − ℚ(R/Z) = Φ_{R} ,

and this completes the proof of the lemma.

4 The subfunctors of kR_{ℚ}

Notation 4.1 If P is a finite p-group, and if V is any finitely generated ℚP -module, let \overline{V} denote the image of V in kR_{ℚ}(P).

The set \{\overline{V} | V ∈ Irr_{ℚ}(P)\} is a k-basis of kR_{ℚ}(P). If u ∈ kR_{ℚ}(P) and V ∈ Irr_{ℚ}(P), let γ(V, u) denote the element of k defined by

u = ∑_{V ∈Irr_{ℚ}(P)} γ(V, u)\overline{V} .

If P has normal p-rank 1, let H_{P} denote the subfunctor of kR_{ℚ} generated by \overline{Φ}_{P} ∈ kR_{ℚ}(P).

This means that H_{P} is the intersection of all subfunctors F of kR_{ℚ} such that F (P) ∋ \overline{Φ}_{P}. Then clearly, for any any finite p-group Q

H_{P} (Q) = \{kR_{ℚ}(φ)(\overline{Φ}_{P}) | φ ∈ Hom_{C}(P, Q)\} .

Equivalently H_{P} (Q) is the k-subspace of kR_{ℚ}(Q) generated by the images of the elements ℚU ⊗_{ℚP} Φ_{P}, where U is a finite Q-set-P.

Lemma 4.2 Let P be a finite p-group. Let V ∈ Irr_{ℚ}(P), and let (T, S) denote a genetic section of P for V.

1. Let W be any ℚP -module. Then γ(V, \overline{W}) is equal to the image \overline{m(V, W)} of the integer m(V, W) in k.

2. Let u ∈ kR_{ℚ}(P). Then γ(V, u) = γ(Φ_{T/S}, Def_{T/S}^{T}Res_{T}^{P} (u)).

Proof : Assertion 1 is obvious. The equality in Assertion 2 is k-linear in u, hence it suffices to check the case u = \overline{W}, for some ℚP -module W. But again by adjunction

⟨V, V ⟩_{P} m(V, W) = ⟨V, W ⟩_{P} = ⟨Ind_{T}^{P} Inf_{T/S}^{T}Φ(T/S), W ⟩_{P}
= ⟨Φ_{T/S}, Def_{T/S}^{T}Res_{T}^{P} W ⟩_{T/S}
= ⟨Φ_{T/S}, Φ_{T/S}⟩_{T/S} m(Φ_{T/S}, Def_{T/S}^{T}Res_{T}^{P} W) .

The equality m(V, W) = m(Φ_{T/S}, Def_{T/S}^{T}Res_{T}^{P} W) follows, since moreover ⟨V, V ⟩_{P} = ⟨Φ_{T/S}, Φ_{T/S}⟩_{T/S}. Now the result follows from Assertion 1.

Lemma 4.3 Let F be a subfunctor of kR_{ℚ}, and P be a finite p-group. Let u ∈ F (P), and V ∈ Irr_{ℚ}(P) such that γ(V, u) ≠ 0. If V has type R, then \overline{Φ}_{R} ∈ F (R).

Proof : Let (T, S) be a genetic section of P for V. Thus T/S ≅ R. Set n_{V} = γ(V, u). One has that n_{V} = γ(Φ_{T/S}, v) by Lemma 4.2, where v = Def_{T/S}^{T}Res_{T}^{P} (u). In particular v ∈ F (T/S) since F is a subfunctor of kR_{ℚ}. By isomorphism, it follows that there is an element w ∈ F (R) such that n_{V} = γ(Φ_{R}, w) ≠ 0.

Now proceed by induction on |R| : if |R| = 1, since kR_{ℚ}(1) is one dimensional, it follows that \overline{Φ}_{1} ∈ F (1).

Otherwise, the element w is equal to

w = n_{V} \overline{Φ}_{R} + ∑_{{W ∈Irr_{ℚ}(R) \atop W ≠Φ_{R}}} m_{W} \overline{W} ,

where m_{W} = γ(W, w). But all the simple ℚR-modules W different from Φ_{R} are of type S, with |S| < |R|, by Proposition 3.7. By induction hypothesis, one has that \overline{Φ}_{S} ∈ F (S) whenever m_{W} ≠ 0. Thus \overline{W} ∈ F (R), since W is obtained from Φ_{R} by inflation followed by induction, and since F is a subfunctor of kR_{ℚ}.

By difference, it follows that n_{V} \overline{Φ}_{R} ∈ F (R), hence \overline{Φ}_{R} ∈ F (R), as was to be shown.

Theorem 4.4 If F is a subfunctor of kR_{ℚ}, then

F = ∑_{P ∈[N_{F}]} H_{P} ,

where N_{F} is the subclass of N consisting of p-groups R for which \overline{Φ}_{R} ∈ F (R).

Proof : Observe first that the inclusion

∑_{P ∈[N_{F}]} H_{P} ⊆ F

is obvious : if \overline{Φ}_{R} ∈ F (R), then F contains the subfunctor H_{R} generated by \overline{Φ}_{R}.

Conversely, let Q be a finite p-group, and u = ∑_{V ∈Irr_{ℚ}(Q)} γ(V, u)\overline{V} be an element of F (Q). Then by Lemma 4.3, if V ∈ Irr_{ℚ}(Q) has type R and if γ(V, u) ≠ 0, one has that \overline{Φ}_{R} ∈ F (R), i.e. R ∈ N_{F}. Now \overline{V} ∈ H_{R}(Q) since V is obtained from Φ_{R} by inflation followed by induction. It follows that u ∈ ∑_{R∈N_{F}} H_{R}(Q), as was to be shown.

Proposition 4.5 Let F be a subclass of N. If P is any element of N such that H_{P} ⊆ ∑_{R∈[F]} H_{R}, then there exists R ∈ F such that H_{P} ⊆ H_{R}.

Proof : Indeed, if H_{P} ⊆ ∑_{R∈[F]} H_{R}, then there is a finite subset S of [F], and for R ∈ S, there is an element φ_{R} ∈ Hom_{C}(R, P), such that

\overline{Φ}_{P} = ∑_{R∈S} kR_{ℚ}(φ_{R})(\overline{Φ}_{R}) .

It follows in particular that there is some element R ∈ S with γ(Φ_{P} , v) ≠ 0 for some element v = kR_{ℚ}(φ_{R})(\overline{Φ}_{R}) of H_{R}(P). Lemma 4.3 shows then that \overline{Φ}_{P} ∈ H_{R}(P), i.e. that H_{P} ⊆ H_{R}.

Notation 4.6 Denote by P ⪯ Q the relation on N defined by

P ⪯ Q ⇔ H_{P} ⊆ H_{Q} .

A subclass A of N is said to be closed if

∀P ∈ N , ∀Q ∈ A, P ⪯ Q ⇒ P ∈ A .

A subset S of [N] is said to be closed if S = [A] for some closed subclass A of N.

Theorem 4.7 The correspondences sending a subfunctor F of kR_{ℚ} to the subset [N_{F}] of [N], and the subset S of [N] to the subfunctor ∑_{R∈S} H_{R}, are

mutual inverse isomorphisms between the lattice of subfunctors of kR_{ℚ} (ordered by inclusion of subfunctors), and the lattice of closed subsets of [N] (ordered by inclusion of subsets).

Proof : First, if F is a subfunctor of kR_{ℚ}, then N_{F} is a closed subclass of N, by definition of N_{F}.

Denote by A the first correspondence, and by B the second one. Then clearly A and B are maps of posets. Theorem 4.4 shows that B ∘ A is equal to the identity, and Proposition 4.5 shows that A∘B is equal to the identity : indeed if S ⊆ [N], and if P ∈ N is such that H_{P} ⊆ ∑_{R∈S} H_{R}, then there exists R ∈ S with H_{P} ⊆ H_{R}. Hence P ∈ S since S is a closed subset of [N]. Thus N_{B(S)} = S.

5 The functors H_{P}

Proposition 5.1 Let P ∈ N.

1. The functor H_{P} has a unique maximal (proper) subfunctor J_{P}, defined

by

J_{P} = ∑_{R} H_{R} ,

where the sum runs through the elements R of [N] for which H_{R} is a proper subfunctor of H_{P}.

2. The quotient functor H_{P} /J_{P} is isomorphic to the simple functor S_{Q,k}, where Q is an element of N of minimal order such that H_{Q} = H_{P}.

3. If Q is any element of N of minimal order such that H_{Q} = H_{P}, then Q is isomorphic to a section of P.

Proof : Let F be a proper subfunctor of H_{P}. Then F is equal to the sum of the functors H_{Q} it contains. Those are all proper subfunctors of H_{P}. Thus F ⊆ J_{P}. And J_{P} is indeed a proper subfunctor of H_{P}, by Proposition 4.5. This proves Assertion 1.

Now the quotient S = H_{P} /J_{P} is a simple functor, isomorphic to a simple functor S_{Q,V}, for some p-group Q and some kOut(Q)-module V : the group Q is of minimal order subject to S(Q) ≠ 0, and then V = S(Q), as kOut(Q)- module.

Let u ∈ H_{P} (Q) − J_{P} (Q), and write u = ∑_{V ∈Irr_{ℚ}(Q)} γ(V, u)\overline{V}. Then if γ(V, u) ≠ 0, and V has type R, Lemma 4.3 shows that \overline{Φ}_{R} ∈ H_{P} (R). If |R| < |Q| then H_{P} (R) = J_{P} (R) by minimality of Q. In this case \overline{V} ∈ J_{P} (Q). Since u ≠ 0, it means that there must exist a V ∈ Irr_{ℚ}(Q) of type Q, and such that γ(V, u) ≠ 0. In other words Q ∈ N, and γ(Φ_{Q}, u) ≠ 0. Moreover H_{P} (Q) = J_{P} (Q) + k\overline{Φ}_{Q}. Now \overline{Φ}_{Q} ∈ H_{P} (Q), hence H_{Q} ⊆ H_{P}. And since \overline{Φ}_{Q} ∉ J_{P} (Q), it follows that H_{Q} is not contained in J_{P}. Hence H_{Q} = H_{P}.

Now if Q′ ∈ N is such that |Q′| < |Q| and H_{Q′} = H_{P}, then also J_{Q′} = J_{P}, hence S ≅ H_{Q′}/J_{Q′}. But S(Q′) = 0 by minimality of Q. In particular \overline{Φ}_{Q′} ∈ J_{Q′}(Q′), hence H_{Q′} ⊆ J_{Q′}, which is not possible since J_{Q′} is a proper subfunctor of H_{Q′}. Thus Q has minimal order such that H_{Q} = H_{P}.

Finally S(Q) = H_{P} (Q)/J_{P} (Q) is one dimensional over k, generated by the image of \overline{Φ}_{Q}. This element is clearly invariant by Out(Q). Hence S(Q) is the trivial module, and S ≅ S_{Q,k}.

Conversely, if R is any element of N such that H_{R} = H_{P}, then also J_{R} = J_{P}, and H_{R}/J_{R} ≅ S_{Q,k}. In particular S_{Q,k}(R) ≠ 0, since otherwise H_{R}(R) = J_{R}(R), and \overline{Φ}_{R} ∈ J_{R}(R), hence H_{R} = J_{R}, a contradiction. By Lemme 7 page 678 of [1], this implies that Q is isomorphic to a section of R (hence also of P). In particular, any two elements R of minimal order such that H_{R} = H_{P} are isomorphic.

Lemma 5.2 Let P and Q be finite p-groups, and let U be a finite Q-set-P. Let V be a simple ℚP -module such that ⟨V, V ⟩_{P} ≥ p, and let W be a simple ℚQ-module.

If ⟨W, W ⟩_{Q} < ⟨V, V ⟩_{P}, then m(W, ℚU ⊗_{ℚP} V) is a multiple of p.

Proof : Indeed by adjunction

⟨W, ℚU ⊗_{ℚP} V ⟩_{Q} = dim_{ℚ} Hom_{ℚQ}(W, ℚU ⊗_{ℚP} V)
= dim_{ℚ} Hom_{ℚP}(ℚU^{op} ⊗_{ℚQ} W, V)
= ⟨ℚU^{op} ⊗_{ℚQ} W, V ⟩_{P}

where U^{op} denotes the opposite biset, i.e. the set U viewed as a P -set-Q by x.u.y(in U^{op}) = y^{−1}ux^{−1}(in U), for x ∈ P, y ∈ Q, u ∈ U.

It follows that

⟨W, W ⟩_{Q} m(W, ℚU ⊗_{ℚP} V) = ⟨V, V ⟩_{P} m(V, ℚU^{op} ⊗_{ℚQ} W) .

But since ⟨V, V ⟩_{P} ≥ p, it follows from Assertion 3 of Proposition 2.5 that there is an integer a > 0 such that

⟨V, V ⟩_{P} = (p − 1)p^{a} .

Now either W is trivial and then

m(W, ℚU ⊗_{ℚP} V) = (p − 1)p^{a} m(V, ℚU^{op} ⊗_{ℚQ} W) ≡ 0 (p)

or W is non-trivial, and there exists an integer b ≥ 0 such that

⟨W, W ⟩_{Q} = (p − 1)p^{b} .

Moreover b < a since ⟨W, W ⟩_{Q} < ⟨V, V ⟩_{P}. Then

m(W, ℚU ⊗_{ℚP} V) = p^{a−b}m(V, ℚU^{op} ⊗_{ℚQ} W) ≡ 0 (p)

as was to be shown.

Notation 5.3 Let m ∈ ℕ. If P is a finite p-group, denote by N_{m}(P) the k-subspace of kR_{ℚ}(P) generated by the elements \overline{V}, for V ∈ Irr_{ℚ}(P), such that ⟨V, V ⟩_{P} ≡ 0 (p^{m}).

Lemma 5.4 With this notation N_{m} is a subfunctor of kR_{ℚ}.

Proof : If m = 0 there is nothing to prove, since N_{0} = kR_{ℚ}. One can suppose m ≥ 1, and the result follows then from Lemma 5.2 : if P and Q are finite p-groups, if φ ∈ Hom_{C}(P, Q), if V ∈ Irr_{ℚ}(P) is such that ⟨V, V ⟩_{P} ≡ 0 (p^{m}), and if W ∈ Irr_{ℚ}(Q) are such that γ(W, kR_{ℚ}(φ)(\overline{V})) ≠ 0, then there is a finite Q-set-P, say U, such that the multiplicity of W in ℚU ⊗_{ℚP} V is not divisible by p. Since ⟨V, V ⟩_{P} ≥ p, it follows from Lemma 5.2 that ⟨W, W ⟩_{Q} ≥ ⟨V, V ⟩_{P}. But there are integers a and b such that ⟨V, V ⟩_{P} = (p − 1)p^{a}, and ⟨W, W ⟩_{Q} = (p − 1)p^{b}. Thus b ≥ a, and a ≥ m if ⟨V, V ⟩_{P} ≡ 0 (p^{m}). Hence ⟨W, W ⟩_{Q} ≡ 0 (p^{m}), as was to be shown.

Proposition 5.5 Let P be a p-group of normal p-rank 1, of order p^{d}.

1. If P is cyclic or generalized quaternion, then

⟨Φ_{P} , Φ_{P} ⟩_{P} = \{ \begin{matrix}1 & if d = 0 \\ (p − 1)p^{d−1} & if d ≥ 1 \end{matrix} .

2. If P is dihedral or semi-dihedral, then

⟨Φ_{P} , Φ_{P} ⟩_{P} = 2^{d−3} .

Proof : Suppose first that P is cyclic or generalized quaternion. If P = 1 the result is trivial. Otherwise the proof of Proposition 3.7 shows that Φ_{P} = V_{1}. Now by Assertion 3 of Proposition 2.5

⟨V_{1}, V_{1}⟩_{P} = (p − 1)\frac{|N_{P} (1) : 1|}{p}|I_{P} (1, 1)| = (p − 1)\frac{|P |}{p} .

Assertion 1 follows.

Similarly, if P is dihedral or semi-dihedral, and if Q is a non-central subgroup of P of order 2, then Φ_{P} = V_{Q}, and

⟨V_{Q}, V_{Q}⟩_{P} = \frac{|N_{P} (Q) : Q|}{2}|I_{P} (Q, Q)| .

Moreover \frac{|N_{P} (Q):Q|}{2} = 1 in this case, since N_{P} (Q) has order 4. And if x ∈ P is such that Q^{x} ∩ N_{P} (Q) ⊆ Q, then either x ∈ N_{P} (Q), or x ∉ N_{P} (N_{P} (Q)), and then Q^{x} ∩ N_{P} (Q) = 1, or equivalently |QxN_{P} (Q)| = 8. Since N_{P} (N_{P} (Q)) has order 8, there are (|P | − 8)/8 such double cosets QxN_{P} (Q). This shows that |I_{P} (Q, Q)| = |P |/8 = 2^{d−3}, and Assertion 2 follows. Note that the argument still holds if P is a dihedral group of order 8 (which is not of normal 2 rank 1) : if Q is a non-central subgroup of P of order 2, then Q is basic, and ⟨V_{Q}, V_{Q}⟩_{P} = 2^{3−3} = 1.

Theorem 5.6 Let P and Q be finite p-groups of normal p-rank 1. Then H_{P} = H_{Q} if and only if one of the following holds :

1. One has that |P | ≤ p and |Q| ≤ p. In this case H_{P} = H_{Q} = kR_{ℚ}.

2. One has |P | ≥ p^{2}, and Q ≅ P.

Proof : If H_{P} = kR_{ℚ}, then in particular H_{P} is not contained in N_{1}, which is a proper subfunctor of kR_{ℚ} (since ⟨Φ_{1}, Φ_{1}⟩_{1} = 1 ≢ 0 (p)). It follows that ⟨Φ_{P} , Φ_{P} ⟩_{P} is not divisible by p. By Proposition 5.5, this can only happen if |P | ≤ p.

Conversely, if |P | ≤ p, then H_{P} = kR_{ℚ} : first suppose that P = 1. Let Q be a finite p-group. If R is a subgroup of Q, then

ℚ(Q/R) = Ind_{R}^{Q}Inf_{R/R}^{R}ℚ .

In other words, if U is the set Q/R, viewed as a Q-set-1, then ℚ(Q/R) = R_{ℚ}(U)(ℚ). Since ℚ = Φ_{1}, and since any element of kR_{ℚ}(Q) is a k-linear combination of elements \overline{ℚ(Q/R)}, it follows that H_{1} = kR_{ℚ}.

Now if P is cyclic of order p, then

Φ_{P} = ℚ(P/1) − ℚ(P/P) .

Restriction to the trivial group gives

Res_{1}^{P} Φ_{P} = pℚ − ℚ = (p − 1)ℚ = (p − 1)Φ_{1} .

Now p − 1 = −1 in k, proving that \overline{Φ}_{1} ∈ H_{P} (1). Thus H_{1} ⊆ H_{P}, and H_{P} = kR_{ℚ}.

Now suppose P has order at least p^{2}. Then ⟨Φ_{P} , Φ_{P} ⟩ ≡ 0 (p), thus H_{P} ⊆ N_{1}. Hence if H_{Q} = H_{P}, then H_{Q} ⊆ N_{1}, hence also |Q| ≥ p^{2}. There are positive integers m and l such that ⟨Φ_{P} , Φ_{P} ⟩_{P} = (p−1)p^{m}, and ⟨Φ_{Q}, Φ_{Q}⟩_{Q} = (p − 1)p^{l}. Thus H_{P} ⊆ N_{m}, and then H_{Q} ⊆ N_{m}. Hence ⟨Φ_{Q}, Φ_{Q}⟩_{Q} ≡ 0 (p^{m}), and l ≥ m. By symmetry l = m.

To prove that Q ≅ P, it suffices to prove that the hypothesis H_{P} = H_{Q} and |P | ≥ p^{2} implies |P | = |Q|. Indeed, in that case, the groups P and Q will both be of minimal order such that H_{P} = H_{Q}. Hence each will be isomorphic to a section of the other, by Assertion 3 of Proposition 5.1.

Suppose then that |Q| < |P |, and that ⟨Φ_{Q}, Φ_{Q}⟩_{Q} = ⟨Φ_{P} , Φ_{P} ⟩_{P}. By Proposition 5.5, this can only happen if p = 2, if P is dihedral or semidihedral of order 2^{d}, and Q is cyclic or generalized quaternion of order 2^{d−2}. The following lemma shows that \overline{Φ}_{Q} ∉ H_{P} (Q) in that case, completing the proof.

Remark 5.7 This proof gives the reason why the cyclic group of order p plays a special rôle in this work : the group C_{p} is the only non trivial element P of N for which the dimension of Φ_{P} is not divisible by p, or equivalently, for which Res_{1}^{P} \overline{Φ}_{P} ≠ 0.

Lemma 5.8 Let P be a dihedral or semi-dihedral group of order 2^{d}, and Q be a cyclic or generalized quaternion group of order 2^{d−2}. Then if U is a finite Q-set-P, the multiplicity of Φ_{Q} in ℚU ⊗_{ℚP} Φ_{P} is even.

Proof : One can suppose that U is a transitive biset, and in this case (Lemme 3 of [1], or Lemma 7.4 of [4]), there are subgroups X of P and Y of Q, there is a normal subgroup R of X, and a normal subgroup R′ of Y, with X/R ≅ Y/R′, such that

ℚU ⊗_{ℚP} Φ_{P} = Ind_{Y}^{Q}Inf_{Y/R′}^{Y}Iso_{X/R}^{Y/R′}Def_{X/R}^{X}Res_{X}^{P}(Φ_{P}) .

Since |Q| < |P |, then |X/R| < |P |, which forces |X| < |P | or |R| ≠ 1. Suppose first that X = P. Then R is a non-trivial normal subgroup of P, and Def_{P/R}^{P}Φ_{P} = 0 by Lemma 3.12 in this case.

Now if X ≠ P, the restriction Res_{X}^{P} factors through the restriction to some maximal subgroup M of P. By Lemma 3.14, one has that Res_{M}^{P}Φ_{P} = 2Φ_{M} if M is dihedral, and Res_{M}^{P}Φ_{P} = Φ_{M} if M is cyclic or generalized quaternion. In the first case, all simple modules have even multiplicity in Res_{M}^{P}Φ_{P}. Thus m(Φ_{Q}, ℚU ⊗_{ℚP} Φ_{P}) is even in that case.

In the second case, one has that ⟨Φ_{M}, Φ_{M}⟩_{M} = 2^{d−2}, since M is cyclic or quaternion of order 2^{d−1}. Now there is a Q-set-M, say V, such that

ℚU ⊗_{ℚP} Φ_{P} = ℚV ⊗_{ℚM} Φ_{M} ,

If m(Φ_{Q}, ℚU ⊗_{ℚP} Φ_{P}) = m(Φ_{Q}, ℚV ⊗_{ℚM}Φ_{M}) is odd, it follows by Lemma 5.2 that

⟨Φ_{Q}, Φ_{Q}⟩ ≥ ⟨Φ_{M}, Φ_{M}⟩_{M} = 2^{d−2} .

This is a contradiction since ⟨Φ_{Q}, Φ_{Q}⟩ = 2^{d−3} when Q is cyclic or generalized quaternion of order 2^{d−2}. This completes the proof of the lemma.

Corollary 5.9 Let P be a finite p-group, and let V be a simple ℚP -module. If (T, S) and (T′, S′) are genetic sections of P for V, then T/S ≅ T′/S′.

Proof : Set R = T/S and R′ = T′/S′. Since the trivial module is the only one of trivial type, one can suppose that V is non-trivial, and that R and R′ are non-trivial. By assumption V ≅ Ind_{T}^{P} Inf_{R}^{T}Φ_{R} and V ≅ Ind_{T′}^{P}Inf_{R′}^{T′}Φ_{R′}. The first isomorphism implies \overline{V} = Ind_{T}^{P} Inf_{R}^{T}\overline{Φ}_{R}, and then the second one gives \overline{Φ}_{R} ∈ H_{R′}(R). Hence H_{R} ⊆ H_{R′}, thus H_{R} = H_{R′} by symmetry. It follows that R ≅ R′, by Theorem 5.6, since R and R′ are non-trivial.

The following corollary is a summary of the previous results :

Corollary 5.10 Let P ∈ N.

1. If |P | ≤ p, then H_{P} = kR_{ℚ} has a unique maximal subfunctor J_{1}, and the quotient kR_{ℚ}/J_{1} is isomorphic to S_{1,k}.

2. If |P | ≥ p^{2}, then H_{P} has a unique maximal subfunctor J_{P}, and the quotient H_{P} /J_{P} is isomorphic to S_{P,k}.

Corollary 5.11 Let F be a subfunctor of kR_{ℚ}, and let F′ be a maximal proper subfunctor of F. Then there exists P ∈ N, with P ≅ C̸_{p}, such that F/F′ ≅ S_{P,k}.

Proof : Indeed since F′ is a proper subfunctor of F, by Theorem 4.4, there must be P ∈ N such that H_{P} ⊆ F but H_{P} ⊈ F′, and one can moreover suppose P ≅ C̸_{p}. Then H_{P} + F′ = F, and F/F′ ≅ H_{P} /(H_{P} ∩ F′) is a simple quotient of H_{P}. Hence it is isomorphic to S_{P,k}.

Theorem 5.12 Let Q be a finite p-group.

1. The dimension of S_{1,k}(Q) is equal to

dim_{k} S_{1,k}(Q) = |{V ∈ Irr_{ℚ}(Q) | V has type 1 or C_{p}}|
= |{V ∈ Irr_{ℚ}(Q) | ⟨V, V ⟩_{Q} ≤ p − 1}|
= |{V ∈ Irr_{ℚ}(Q) | ⟨V, V ⟩_{Q} ≢ 0 (p)}|

2. If P ∈ N and |P | ≥ p^{2}, then

dim_{k} S_{P,k}(Q) = |{V ∈ Irr_{ℚ}(Q) | V has type P }| .

Proof : If P ∈ N and |P | ≠ p, then H_{P} /J_{P} ≅ S_{P,k} by Corollary 5.10. Let u ∈ H_{P} (Q). Then

u = ∑_{V ∈Irr_{ℚ}(Q)} γ(V, u)\overline{V} .

If V ∈ Irr_{ℚ}(Q) has type R, and if γ(V, u) ≠ 0, then by Lemma 4.3 one has that \overline{Φ}_{R} ∈ H_{P} (R), thus H_{R} ⊆ H_{P}. If the inclusion is proper, then H_{R} ⊆ J_{P}, and \overline{Φ}_{R} ∈ J_{P} (R). It follows that \overline{V} ∈ J_{P} (Q).

Hence H_{P} (Q) is generated by the elements \overline{V}, where V is of type R ⪯ P, and moreover \overline{V} ∈ J_{P} (Q) if H_{R} is properly contained in H_{P}. The quotient H_{P} (Q)/J_{P} (Q) is then generated by the images of the elements \overline{V}, for V ∈ Irr_{ℚ}(Q) of type R such that H_{R} = H_{P}. By Corollary 5.10, if P = 1, this implies |R| ≤ p, and if |P | ≥ p^{2}, this implies R ≅ P.

Now suppose that u ∈ H_{P} (Q) is a linear combination of elements \overline{V}, for V ∈ Irr_{ℚ}(P), of type R such that H_{R} = H_{P}, and that u ∈ J_{P} (Q). Then if γ(V, u) ≠ 0 for some V ∈ Irr_{ℚ}(Q), Lemma 4.3 shows that \overline{Φ}_{R} ∈ J_{P} (R), and then H_{R} = H_{P} ⊆ J_{P}, a contradiction. It follows that u = 0, and the images of the elements \overline{V}, for V ∈ Irr_{ℚ}(P), of type R such that H_{R} = H_{P}, form a k-basis of H_{P} (Q)/J_{P} (Q). The theorem follows then from Proposition 2.5 and Proposition 5.5.

Remark 5.13 If p = 2, Theorem 5.12 implies that the k-dimension of S_{1,k}(Q) is equal to the number of absolutely irreducible rational representations of Q.

Remark 5.14 I. Bourizk ([5] Proposition 4 and Proposition 6) has given upper and lower bounds for dim_{k} S_{P,k}(Q), when P and Q are finite p-groups with |P | ≠ p. One can check that these bounds are compatible with the values given here, in the case P ∈ N : for example, if P has order at least p^{2}, the lower bound in Proposition 4 of [5] is equal to the number l of normal subgroups R of Q such that Q/R ≅ P. If P ∈ N, then each of these quotients has a unique faithful rational irreducible representation, and inflating to P gives at least l non-isomorphic irreducible rational representations of Q of type P. The other verifications are similar.

6 The lattice of subfunctors of kR_{ℚ}

The following is the first part of Theorem 1.3 :

Theorem 6.1 If k is a field of odd characteristic p, then the functor kR_{ℚ} is a uniserial object of F_{p,k}. If F_{0} = kR_{ℚ} ⊃ F_{1} ⊃ F_{2} ⊃ … ⊃ F_{n} ⊃ … is the set of its non-zero subfunctors, then

F_{0}/F_{1} ≅ S_{1,k} F_{i}/F_{i+1} ≅ S_{C_{p^{i+1}},k} for i ≥ 1 ,

where C_{p^{i+1}} denotes a cyclic group of order p^{i+1}.

Proof : Indeed, if p ≠ 2, then N just consists of cyclic groups. If P is cyclic of order p^{k} ≥ p^{2}, then Φ_{P} = ℚ(P/1) − ℚ(P/Z), where Z is the only subgroup of order p of P. If Q is the only subgroup of P of index p, then Φ_{P} = Ind_{Q}^{P}Φ_{Q}. This shows that \overline{Φ}_{P} ∈ H_{Q}(P), i.e. that H_{P} ⊆ H_{Q}. By Corollary 5.10, the inclusion is proper.

Now the set of subfunctors H_{C_{p^{m}}}, for m ≥ 2, is linearly ordered by inclusion. Moreover recall that H_{1} = H_{C_{p}} = kR_{ℚ}. Proposition 5.1 shows that J_{1} = H_{C_{p^{2}}}, and that J_{P} = H_{C_{p^{m+1}}} if P is cyclic of order p^{m} ≥ p^{2}. Since J_{P} is the unique maximal subfunctor of H_{P}, it follows that kR_{ℚ} is uniserial, with filtration

kR_{ℚ} ⊃ H_{C_{p^{2}}} ⊃ H_{C_{p^{3}}} ⊃ … ⊃ H_{C_{p^{m}}} ⊃ ….

Moreover the quotient kR_{ℚ}/H_{C_{p^{2}}} is isomorphic to S_{1,k}, and the quotient H_{C_{p^{m}}}/H_{C_{p^{m+1}}} is isomorphic to S_{C_{p^{m}}}, for m ≥ 2.

Theorem 6.2 Suppose p = 2, and consider the following graph structure on [N_{1}] :

[figure: •C_{4}; •SD_{16}; •D_{16}; •Q_{8}; •C_{8}; •SD_{32}; •D_{32}; •Q_{16}; •C_{16}; •SD_{64}; •D_{64}; •Q_{32}; •C_{32}; •SD_{128}; •D_{128}; …; …; …; …]

Then for P and Q in [N_{1}], one has that P ⪯ Q if and only if there is an (oriented) path from Q to P in [N_{1}].

Proof : First check that if [figure: R; P] is an arrow in [N_{1}], then P ⪯ R. There are several cases :

• [figure: X_{2^{n}}; X_{2^{n+1}}] , where X is one of Q, C, or D : in this case X_{2^{n}} is a subgroup of X_{2^{n+1}}, and Φ_{X_{2^{n+1}}} = Ind_{X_{2^{n}}}^{X_{2^{n+1}}}Φ_{X_{2^{n}}}, thus X_{2^{n+1}} ⪯ X_{2^{n}}.

• [figure: C_{2^{n}}; Q_{2^{n+1}}] : here again Φ_{Q_{2^{n+1}}} = Ind_{C_{2^{n}}}^{Q_{2^{n+1}}}Φ_{C_{2^{n}}}, and it follows that Q_{2^{n+1}} ⪯ C_{2^{n}}.

• [figure: SD_{2^{n}}; X_{2^{n−1}}] , where X is Q or C : here X_{2^{n−1}} is a subgroup of SD_{2^{n}}, and Res_{X_{2^{n−1}}}^{SD_{2^{n}}}Φ_{SD_{2^{n}}} = Φ_{X_{2^{n−1}}}, by Lemma 3.14. It follows that X_{2^{n−1}} ⪯ SD_{2^{n}}.

• [figure: D_{2^{n}}; C_{2^{n−1}}] : here again Res_{C_{2^{n−1}}}^{D_{2^{n}}}Φ_{D_{2^{n}}} = Φ_{C_{2^{n−1}}}, and it follows that C_{2^{n−1}} ⪯ D_{2^{n}}.

• [figure: D_{2^{n}}; SD_{2^{n+1}}] : here again Φ_{SD_{2^{n+1}}} = Ind_{SD_{2^{n+1}}}^{D_{2^{n}}}Φ_{D_{2^{n}}}, and it follows that SD_{2^{n+1}} ⪯ D_{2^{n}}.

Conversely, suppose that P and R are in [N_{1}], and that P ⪯ R. Then there exists φ ∈ Hom_{C}(R, P) such that \overline{Φ}_{P} = kR_{ℚ}(φ)(\overline{Φ}_{R}). Hence there is a finite P -set-Q, say U, such that m(Φ_{P} , ℚU ⊗_{ℚR} Φ_{R}) is odd. One can moreover suppose that U is a transitive biset. In other words, there are subgroups Y \underline{◁} X ⊆ P and T \underline{◁} Z ⊆ R, with X/Y ≅ Z/T, such that ℚU ⊗_{R} Φ_{R} is equal to

Ψ = Ind_{X}^{P} ∘ Inf_{X/Y}^{X} ∘ Iso_{Z/T}^{X/Y} ∘ Def_{Z/T}^{Z} ∘ Res_{Z}^{R}Φ_{R} .

Since m(Φ_{P} , Ψ) is odd, it follows in particular that Res_{Z}^{R}Φ_{R} ∉ 2R_{ℚ}(Z). But it follows easily from Lemma 3.14 and from Remark 3.3 that if R ∈ N_{1} and |R : Z| ≥ 4, then Res_{Z}^{R} ∈ 2R_{ℚ}(Z). And if |R : Z| = 2, the only case where Res_{Z}^{R} ∉ 2R_{ℚ}(Z) is when R is dihedral or semi-dihedral, and Z is cyclic or generalized quaternion.

In this case Res_{Z}^{R}Φ_{R} = Φ_{Z}. Then Z ⪯ R, and P ⪯ Z, since m(Φ_{P} , Ψ) is odd and Ψ = Ind_{X}^{P}Inf_{X/Y}^{X} Iso_{Z/T}^{X/Y} Def_{Z/T}^{Z} Φ_{Z}. Since moreover [figure: R; Z] in [N_{1}], it suffices to consider the case Z = R.

Then T = 1 by Lemma 3.12, and one can suppose R = X/Y. Moreover

⟨Φ_{P} , Φ_{P} ⟩_{P} m(Φ_{P} , Ψ) = ⟨Φ_{R}, Φ_{R}⟩_{R} m(Φ_{R}, Def_{R}^{X}Res_{X}^{P}Φ_{P}) .  (6.3)

Now X is a subgroup of P ∈ N, and X has a quotient R in N_{1}. By Remark 3.3, it follows that X ∈ N_{1}. Now by Lemma 3.14 and Lemma 3.12, the right hand side of equation 6.3 is zero if Def_{R}^{X} is a proper deflation, i.e. if R ≅ X̸.

Hence one can suppose that R is a subgroup of P, such that m(Φ_{P} , Ind_{R}^{P}Φ_{R}) is odd. There are several cases :

• If P is cyclic or generalized quaternion, then there is indeed a path from R to P in [N_{1}].

• If P is dihedral, then there is a path from R to P in [N_{1}] if R is dihedral. If R is cyclic, then R is contained in a cyclic subgroup C of index 2 in P. Since Φ_{C} = Ind_{R}^{C}Φ_{R}, it follows that m(Φ_{P} , Ind_{C}^{P}Φ_{C}) is odd, and then P ⪯ C. This cannot occur, since C ⪯ P in this case.

• Finally, if P is semi-dihedral, then there is a path from R to P in [N_{1}] if R is cyclic or generalized quaternion. And the case where R is dihedral cannot occur, since R ⪯ P in this case.

Definition 6.4 A subset S of [N_{1}] is called closed if for any arrow [figure: R; P] in [N_{1}] with R ∈ S, one has that P ∈ S.

Corollary 6.5 The lattice of proper subfunctors of kR_{ℚ} is isomorphic to the lattice of closed subsets of the graph [N_{1}], ordered by inclusion of subsets.

Proof : This follows from Theorem 4.7.

Remark 6.6 This shows in particular that the functor kR_{ℚ} is not a noetherian object of F_{2,k}, since the sequence S_{n} = ∑_{l=4}^{l=n} H_{SD_{2^{l}}}, for n ≥ 4, is an infinitely increasing sequence of subobjects of kR_{ℚ}.

Corollary 6.7 The functor H_{Q_{8}} is a uniserial object of F_{2,k}. The set of its non-zero subfunctors is

H_{Q_{8}} ⊃ H_{Q_{16}} ⊃ … ⊃ H_{Q_{2^{n}}} ⊃ ….

Moreover H_{Q_{2^{n}}}/H_{Q_{2^{n+1}}} ≅ S_{Q_{2^{n}},k}, for n ≥ 3.

Proof : Indeed, if Q ∈ N_{1} and Q ⪯ Q_{8}, then Q is generalized quaternion. Thus H_{Q_{2^{n+1}}} = J_{Q_{2^{n}}}.

Remark 6.8 This result was Conjecture 2 in [5], since (with notation of [5]) one has that ε_{Q_{8}} = \overline{Φ}_{Q_{8}}.

The following theorem implies part 2 of Theorem 1.3 :

Theorem 6.9 Suppose that p = 2. If l ∈ ℕ, then

N_{l} = \{ \begin{matrix}kR_{ℚ} & if l = 0 \\ H_{C_{4}} + H_{SD_{16}} + H_{D_{16}} & if l = 1 \\ H_{Q_{2^{l+1}}} + H_{C_{2^{l+1}}} + H_{SD_{2^{l+3}}} + H_{D_{2^{l+3}}} & if l ≥ 2 \end{matrix}

Moreover ∩_{l∈ℕ}N_{l} = 0, and

N_{l}/N_{l+1} ≅ \{ \begin{matrix}S_{1,k} & if l = 0 \\ S_{C_{4},k} ⊕ S_{SD_{16},k} ⊕ S_{D_{16},k} & if l = 1 \\ S_{Q_{2^{l+1}}} ⊕ S_{C_{2^{l+1}}} ⊕ S_{SD_{2^{l+3}}} ⊕ S_{D_{2^{l+3}}} & if l ≥ 2 \end{matrix} .

Proof : Since any subfunctor of kR_{ℚ} is equal to the sum of the subfunctors H_{P} it contains, it follows from Theorem 6.2 that

J_{1} = H_{C_{4}} + H_{SD_{16}} + H_{D_{16}} .

Moreover if P ∈ N_{1}, then ⟨Φ_{P} , Φ_{P} ⟩_{P} is even by Proposition 5.5. It follows that Φ_{P} ∈ N_{1}(P), hence that H_{P} ⊆ N_{1}. Thus J_{1} ⊆ N_{1}, and J_{1} = N_{1} since N_{1} is a proper subfunctor of kR_{ℚ}. In particular, it follows that N_{0}/N_{1} ≅ S_{1,k}.

Now if m ∈ ℕ, with m ≥ 2, and P is one of Q_{2^{m+1}}, C_{2^{m+1}}, SD_{2^{m+3}} or D_{2^{m+3}}, then ⟨Φ_{P} , Φ_{P} ⟩_{P} = 2^{m}, thus H_{P} ⊆ N_{l}, for l ∈ ℕ, if and only if m ≥ l. Hence

N_{l} = ∑_{m≥l} H_{Q_{2^{m+1}}} + ∑_{m≥l} H_{C_{2^{m+1}}} + ∑_{m≥l} H_{SD_{2^{m+3}}} + ∑_{m≥l} H_{D_{2^{m+3}}}
= H_{Q_{2^{l+1}}} + H_{C_{2^{l+1}}} + H_{SD_{2^{l+3}}} + H_{D_{2^{l+3}}} ,

since moreover for m ≥ l

H_{Q_{2^{m+1}}} ⊆ H_{Q_{2^{l+1}}}  H_{C_{2^{m+1}}} ⊆ H_{C_{2^{l+1}}}  H_{SD_{2^{m+3}}} + H_{D_{2^{m+3}}} ⊆ H_{D_{2^{l+3}}} .

Let I = ∩_{l∈ℕ}N_{l}. Then I cannot contain any functor H_{P}. Thus I = 0.

Now for l ≥ 2 the functor H_{P} is a proper subfunctor of H_{Q_{2^{l+1}}} if and only if P ≅ Q_{2^{m}}, with m ≥ l + 2. Hence J_{Q_{2^{l+1}}} = H_{Q_{2^{l+2}}}, whence J_{Q_{2^{l+1}}} ⊆ N_{l+1}.

Similarly if l ≥ 1, the functor H_{P} is a proper subfunctor of H_{C_{2^{l+1}}} if and only if P ≅ Q_{2^{m}} or P ≅ C_{2^{m}}, with m ≥ l + 2. Thus J_{C_{2^{l+1}}} = H_{Q_{2^{l+2}}} + H_{C_{2^{l+2}}}, whence J_{C_{2^{l+1}}} ⊆ N_{l+1}.

Again the functor H_{P} is a proper subfunctor of H_{SD_{2^{l+3}}} if and only if P ≅ Q_{2^{m}} or P ≅ C_{2^{m}}, with m ≥ l + 2. Thus J_{SD_{2^{l+3}}} = H_{Q_{2^{l+2}}} + H_{C_{2^{l+2}}}, and J_{SD_{2^{l+3}}} ⊆ N_{l+1}.

Finally H_{P} is a proper subfunctor of H_{D_{2^{l+3}}} if and only if P ≅ Q_{2^{m}}, with m ≥ l + 3, or P ≅ C_{2^{m}}, with m ≥ l + 2, or P ≅ SD_{2^{m}}, with m ≥ l + 4, or P ≅ D_{2^{m}}, with m ≥ l + 4. Hence J_{D_{2^{l+3}}} = H_{C_{2^{l+2}}} + H_{SD_{2^{l+4}}} + H_{D_{2^{l+4}}}, and J_{D_{2^{l+3}}} ⊆ N_{l+1}.

Let σ_{1} denote the canonical epimorphism

H_{C_{4}} ⊕ H_{SD_{16}} ⊕ H_{D_{16}} → H_{C_{4}} + H_{SD_{16}} + H_{D_{16}} = N_{1} ,

and for l ≥ 2, and let σ_{l} denote the canonical epimorphism

H_{Q_{2^{l+1}}} ⊕ H_{C_{2^{l+1}}} ⊕ H_{SD_{2^{l+3}}} ⊕ H_{D_{2^{l+3}}} → N_{l} ,

where N_{l} = H_{Q_{2^{l+1}}} + H_{C_{2^{l+1}}} + H_{SD_{2^{l+3}}} + H_{D_{2^{l+3}}}. The above remarks show that

σ_{1}(J_{C_{4}} ⊕ J_{SD_{16}} ⊕ J_{D_{16}}) ⊆ N_{2}

and that

σ_{l}(J_{Q_{2^{l+1}}} ⊕ J_{C_{2^{l+1}}} ⊕ J_{SD_{2^{l+3}}} ⊕ J_{D_{2^{l+3}}}) ⊆ N_{l+1}

for l ≥ 2. Let π_{l} : N_{l} → N_{l}/N_{l+1} denote the canonical projection. It follows that π_{1} ∘ σ_{1} factors through an epimorphism

\overline{σ}_{1} : S_{C_{4},k} ⊕ S_{SD_{16},k} ⊕ S_{D_{16},k} → N_{1}/N_{2}

and that π_{l} ∘ σ_{l} factors through an epimorphism

\overline{σ}_{l} : S_{Q_{2^{l+1}}} ⊕ S_{C_{2^{l+1}}} ⊕ S_{SD_{2^{l+3}}} ⊕ S_{D_{2^{l+3}}} → N_{l}/N_{l+1}

for l ≥ 2.

Let K_{l} = σ_{l}^{−1}(N_{l+1}) denote the kernel of π_{l} ∘ σ_{l}. Then

⊕_{P ∈E_{l}} J_{P} ⊆ K_{l} ⊆ ⊕_{P ∈E_{l}} H_{P} ,

where

E_{l} = \{ \begin{matrix}\{C_{4}, SD_{16}, D_{16}\} & if l = 1 \\ \{Q_{2^{l+1}}, C_{2^{l+1}}, SD_{2^{l+3}}, D_{2^{l+3}}\} & if l ≥ 2 \end{matrix} .

Fix P_{0} ∈ E_{l}. Then

J_{P_{0}} ⊆ H_{P_{0}} ∩ (K_{l} + ⊕_{P ∈E_{l}−{P_{0}}} H_{P}) ⊆ H_{P_{0}} .

Since J_{P_{0}} is maximal in H_{P_{0}} it follows that H_{P_{0}} ∩(K_{l} + ⊕_{P ∈E_{l}−{P_{0}}} H_{P}) is equal

to J_{P_{0}} or to H_{P_{0}}. In the latter case H_{P_{0}} ⊆ K_{l} + ⊕_{P ∈E_{l}−{P_{0}}} H_{P}, and taking images by σ_{l} gives

H_{P_{0}} ⊆ σ_{l}(K_{l} + ⊕_{P ∈E_{l}−{P_{0}}} H_{P}) = N_{l+1} + ∑_{P ∈E_{l}−{P_{0}}} H_{P} .

By Theorem 4.4 and Lemma 4.5, it follows that H_{P_{0}} is contained in N_{l+1}, or in H_{P}, for some P ∈ E_{l} − {P_{0}}. Both are impossible, thus

H_{P_{0}} ∩ (K_{l} + ⊕_{P ∈E_{l}−{P_{0}}} H_{P}) = J_{P_{0}} .

This holds for any P_{0} ∈ E_{l}, and it follows that

K_{l} = ⊕_{P ∈E_{l}} J_{P} .

Hence the epimorphisms \overline{σ}_{l} are isomorphisms.

7 Basic subgroups and genetic sections

Recall the following proposition from [1]

Proposition 7.1 ([1] Proposition 16 page 717) Let P and Q be finite pgroups. Let B_{Q}(P) be the free ℤ-module with basis the set of conjugacy classes of pairs (T, S) of subgroups of P with S \underline{◁} T and T/S ≅ Q. If (T, S) and (V, U) are such pairs, set

⟨(T, S) | (V, U)⟩_{ℤ} = |{x ∈ T \P/V | (T, S) (^{x}V,^{x} U)}|

and denote by ⟨(T, S) | (V, U)⟩_{k} the image of ⟨(T, S) | (V, U)⟩_{ℤ} in k. Then there is an isomorphism of k-vector spaces

S_{Q,k}(P) ≅ kB_{Q}(P)/Rad⟨ | ⟩_{k} ,

where kB_{Q}(P) = k ⊗ B_{Q}(P), and Rad⟨ | ⟩_{k} is the radical of the k-valued bilinear form ⟨ | ⟩_{k} on kB_{Q}(P) = k ⊗ B_{Q}(P). In particular, the dimension of S_{Q,k}(P) is equal to the rank of the form ⟨ | ⟩_{k} on kB_{Q}(P), and this bilinear form induces a non-degenerate bilinear form on S_{Q,k}(P), still denoted by ⟨ | ⟩_{k}.

This section describes the relations between basic subgroups and genetic sections, and gives an interpretation of the above proposition in these terms.

Lemma 7.2 Let P ∈ N. If Q is a basic subgroup of P such that V_{Q} ≅ Φ_{P}, then

<I_{P} (Q, Q)> = P ,  ∩_{x∈P} ^{x}Q = 1 .

Proof : There are two cases :

• If P is cyclic or generalized quaternion, then Q = 1, and I_{P} (Q, Q) is equal to P. Then both equalities are obvious.

• If P is dihedral or semi-dihedral, then Q is non-central of order 2. The second equality is clear. For the first one, observe that in this case

I_{P} (Q, Q) = N_{P} (Q) ∪ (P − N_{P} (N_{P} (Q)))

(as in the proof of Proposition 5.5), and that this set generates P.

Definition 7.3 Let P be any finite p-group. If Q is a basic subgroup of P set

T_{Q} = <I_{P} (Q, Q)> , S_{Q} = ∩_{x∈T_{Q}} ^{x}Q .

The basic subgroup Q of P is called an origin (in P) if T_{Q}/S_{Q} has normal p-rank 1. In this case if V is a simple ℚP -module, and if V ≅ V_{Q}, then Q is called an origin of V (or V is said to have origin Q).

Proposition 7.4 Let P be a finite p-group. Then :

1. If V is a simple ℚP -module, let (T, S) be a genetic section of P for V. If Q/S is a basic subgroup of T/S such that

Q/S ∩ Z(T/S) = 1 ,

then Q is an origin of V in P, and (T_{Q}, S_{Q}) = (T, S).

2. If Q is an origin in P, then (T_{Q}, S_{Q}) is a genetic section of V_{Q}. Moreover Q/S_{Q} is a basic subgroup of T_{Q}/S_{Q}, and

Q/S_{Q} ∩ Z(T/S_{Q}) = 1 .

Proof : Let V be a simple ℚP -module. Assertion 1 is trivial if V ≅ ℚ, since (P, P) is the only genetic section of P for V in this case. Hence one can suppose that V is non-trivial. If (T, S) is a genetic section of P for V, then

V ≅ Ind_{T}^{P} Inf_{T/S}^{T}Φ_{T/S}

Let \overline{Q} = Q/S be a basic subgroup of \overline{T} = T/S such that \overline{Q}∩Z(\overline{T}) = 1. This means that the simple ℚ\overline{T} -module corresponding to \overline{Q} is faithful, hence it is isomorphic to Φ_{\overline{T}}. In other words

Φ_{\overline{T}} ≅ Ind\overline{T} ; \overline{R}Inf\overline{R} ; \overline{R}/\overline{Q}Ω_{\overline{R}/\overline{Q}} ,

where \overline{R}/\overline{Q} is the unique subgroup of order p of N_{\overline{T}} (\overline{Q})/\overline{Q}. Then

V ≅ Ind_{R}^{P}Inf_{R/Q}^{R}Ω_{R/Q} .

Since V is irreducible, it follows that Q is a basic subgroup of P, that R = Q̃, and that V ≅ V_{Q}. Now by Proposition 2.5

⟨V, V ⟩_{P} = (p − 1)|{x ∈ Q\P/Q̃ | Q^{x} ∩ Q̃ ⊆ Q}|

But similarly, the group T has an irreducible rational module

V′ = Inf_{T/S}^{T}Φ_{\overline{T}} ≅ IndT ; Q̃InfQ̃ ; Q̃/QΩ_{Q̃/Q} ,

thus Q is a basic subgroup of T and

⟨V′, V′⟩_{T} = (p − 1)|{x ∈ Q\T/Q̃ | Q^{x} ∩ Q̃ ⊆ Q}|

Since ⟨V, V ⟩_{P} = ⟨Φ_{\overline{T}} , Φ_{\overline{T}} ⟩_{\overline{T}} = ⟨V′, V′⟩_{T}, it follows that

|{x ∈ Q\P/Q̃ | Q^{x} ∩ Q̃ ⊆ Q}| = |{x ∈ Q\T/Q̃ | Q^{x} ∩ Q̃ ⊆ Q}| .

In other words I_{P} (Q, Q) = I_{T} (Q, Q). Now clearly

I_{\overline{T}} (\overline{Q}, \overline{Q}) = \{xS | x ∈ I_{P} (Q, Q)\} ,

and \overline{T} is generated by I_{\overline{T}} (\overline{Q}, \overline{Q}) by Lemma 7.2. Hence T is generated by I_{P} (Q, Q), i.e. T = T_{Q}, and moreover

S = ∩_{x∈T} ^{x}Q ,

i.e. S = S_{Q}. Thus Q is an origin of V in P, and (T_{Q}, S_{Q}) = (T, S). This proves Assertion 1.

Conversely, let Q be an origin in P. If Q = P, then V_{Q} = ℚ, and (T_{Q}, S_{Q}) = (P, P). Assertion 2 is trivial in this case, and one can suppose that Q is a proper subgroup of P. In particular Q is a proper basic subgroup of P, and T_{Q} ⊇ N_{P} (Q) ⊇ Q̃. Thus

V_{Q} ≅ Ind_{T_{Q}}^{P}Inf_{T_{Q}/S_{Q}}^{T_{Q}}W_{Q} ,

where

W_{Q} = Ind_{Q̃/S_{Q}}^{T_{Q}/S_{Q}}Inf_{Q̃/Q}^{Q̃/S_{Q}}Ω_{Q̃/Q} .

Now the intersection of conjugates of Q/S_{Q} in T_{Q}/S_{Q} is equal to S_{Q}/S_{Q} = 1. This is equivalent to saying that Q/S_{Q}∩Z(T_{Q}/S_{Q}) = 1. Now W_{Q} is a faithful ℚ(T_{Q}/S_{Q})-module, and W_{Q} is irreducible, since V_{Q} is an irreducible ℚP - module. It follows from Proposition 2.2 that \overline{Q} = Q/S_{Q} is a basic subgroup of \overline{T} = T_{Q}/S_{Q}, and that { \atop Q/S_{Q}} = Q̃/S_{Q} = \overline{Q̃}. Now by Proposition 2.5

⟨W_{Q}, W_{Q}⟩_{\overline{T}} = (p − 1)|\{xS_{Q} ∈ \overline{Q}\\overline{T} /\overline{Q̃} | \overline{Q}^{x} ∩ \overline{Q̃} ⊆ \overline{Q}\}| .

Now the map

θ : \{x ∈ Q\T_{Q}/Q̃ | Q^{x} ∩ Q̃ ⊆ Q\} → \{xS_{Q} ∈ \overline{Q}\\overline{T} /\overline{Q̃} | \overline{Q}^{x} ∩ \overline{Q̃} ⊆ \overline{Q}\}

defined by θ(x) = xS_{Q} is one to one. Hence

⟨W_{Q}, W_{Q}⟩_{\overline{T}} = (p − 1)|\{x ∈ Q\T_{Q}/Q̃ | Q^{x} ∩ Q̃ ⊆ Q\}| .

But again by Proposition 2.5

⟨V_{Q}, V_{Q}⟩_{P} = (p − 1)|{x ∈ Q\P/Q̃ | Q^{x} ∩ Q̃ ⊆ Q}| .

Now if x ∈ P and Q^{x} ∩ Q̃ ⊆ Q, then x ∈ I_{P} (Q, Q) ⊆ T_{Q}. Thus

⟨V_{Q}, V_{Q}⟩_{P} = (p − 1)|\{x ∈ Q\T_{Q}/Q̃ | Q^{x} ∩ Q̃ ⊆ Q\}| = ⟨W_{Q}, W_{Q}⟩_{\overline{T}} ,

showing that (T_{Q}, S_{Q}) is a genetic section of P for V_{Q}.

Corollary 7.5 1. If V ∈ Irr_{ℚ}(P), then there exists an origin for V in P. In other words, each equivalence class of B_{P} for ≑_{P} contains an origin.

2. If Q and R are origins in P, and if Q ≑_{P} R, then T_{Q}/S_{Q} ≅ T_{R}/S_{R}.

Proof : Each assertion follows from the corresponding assertion of Proposition 7.4 : for Assertion 1, by Theorem 3.4, there exists a genetic section (T, S) of P for V, and moreover T/S always has a basic subgroup Q/S intersecting the center Z(T/S) trivially, by Proposition 3.7. For Assertion 2, if Q and R are origins in P, and if Q ≑_{P} R, then V_{Q} ≅ V_{R}. Thus (T_{Q}, S_{Q}) and (T_{R}, S_{R}) are both genetic sections of P for V_{Q}, and Assertion 2 follows from Corollary 5.9.

Corollary 7.6 Let Q be an origin in P.

1. If T_{Q}/S_{Q} is cyclic or generalized quaternion, then Q = S_{Q}, and

T_{Q} = N_{P} (Q) = I_{P} (Q, Q) .

2. If T_{Q}/S_{Q} is dihedral or semi-dihedral, then N_{P} (Q)/Q has order 2. In other words N_{P} (Q) = Q̃.

Proof : Suppose first that T_{Q}/S_{Q} is cyclic or generalized quaternion. Then Q = S_{Q}, since Q/S_{Q} intersects the center of T_{Q}/S_{Q} trivially. Now always N_{P} (Q) ⊆ I_{P} (Q, Q) ⊆ T_{Q}. Moreover T_{Q} normalizes S_{Q} = Q, thus T_{Q} ⊆ N_{P} (Q).

And if T_{Q}/S_{Q} is dihedral or semi-dihedral, then N_{P} (Q) = N_{T_{Q}}(Q), and N_{T_{Q}}(Q)/Q ≅ N_{T_{Q}/S_{Q}}(Q/S_{Q}) ≅ C_{2}. Hence N_{P} (Q)/Q ≅ C_{2}, or equivalently N_{P} (Q) = Q̃.

Remark 7.7 It may happen that a basic subgroup is not an origin : for an example, consider the central product P = D_{16} ∗ C_{4} of order 32, and the subgroup Q which is the image in P of a non-central subgroup of D_{16} of order 2. Then Q is basic in P, but T_{Q} = P and S_{Q} = 1, thus T_{Q}/S_{Q} ≅ P is not in N.

But if R is the subgroup of P generated by ab, where a is an element of order 4 in D_{16} and b is a generator of C_{4}, one can check that R ≑_{P} Q, and that R is an origin in P. It should be noted in that case that the groups N_{P} (Q)/Q and N_{P} (R)/R are not isomorphic : both are cyclic, but the first one has order 4, and the second one has order 8.

This latter point is general : suppose that V is a non-trivial irreducible ℚP -module for some finite p-group P, and that the type of V is cyclic or generalized quaternion. Let Q be a basic subgroup for V of P, and R be an origin of V in P. Then (T, S) = (N_{P} (R), R) is a genetic section of P for V, by Corollary 7.6, thus V ≅ V (T, S), and in particular

⟨V, V ⟩_{P} = ⟨Φ_{T/S}, Φ_{T/S}⟩_{T/S} = (1 − \frac{1}{p})|T : S| ,

by Proposition 5.5. On the other hand, the group U = N_{P} (Q)/Q is cyclic or generalized quaternion, since Q is basic, and it is easy to see that V ≅ Ind_{N_{P} (Q)}^{P}Inf_{U}^{N_{P} (Q)}Φ_{U}, thus

⟨V, V ⟩_{P} ≥ ⟨Φ_{U}, Φ_{U}⟩_{U} = (1 − \frac{1}{p})|U| ,

with equality if and only if the section (N_{P} (Q), Q) is a genetic section of P for V. Thus |N_{P} (Q) : Q| = |U| ≤ |T : S| = |N_{P} (R) : R|, with equality if and only if Q is also an origin of V in P.

Lemma 7.8 Let P, Q ∈ N, and let R be a finite p-group. Let U be a finite R-set-P and V be a finite R-set-Q such that ⟨ℚU ⊗_{ℚP} Φ_{P} , ℚV ⊗_{ℚQ}Φ_{Q}⟩_{R} ≠ 0. Then

⟨ℚU ⊗_{ℚP} Φ_{P} , ℚV ⊗_{ℚQ} Φ_{Q}⟩_{R}^{2} ≥ ⟨Φ_{P} , Φ_{P} ⟩_{P} ⟨Φ_{Q}, Φ_{Q}⟩_{Q} .

If equality holds, then ℚU ⊗_{ℚP} Φ_{P} and ℚV ⊗_{ℚQ}Φ_{Q} have a unique irreducible common direct summand W. Moreover if W is of type S, then H_{S} = H_{P} = H_{Q}, thus either P, Q and S are of order at most p, or P ≅ Q ≅ S.

Proof : Let W ∈ Irr_{ℚ}(R), and set

m_{W} = m(W, ℚU ⊗_{ℚP} Φ_{P}) m′_{W} = m(Φ_{P} , ℚU^{op} ⊗_{ℚR} W)
n_{W} = m(W, ℚV ⊗_{ℚQ} Φ_{Q}) n′_{W} = m(Φ_{P} , ℚV^{op} ⊗_{ℚR} W) .

Then m_{W}, m′_{W}, n_{W} and n′_{W} are non-negative integers, and

ℚU ⊗_{ℚP} Φ_{P} ≅ ⨁_{W ∈Irr_{ℚ}(R)} m_{W} W
ℚV ⊗_{ℚQ} Φ_{Q} ≅ ⨁_{W ∈Irr_{ℚ}(R)} n_{W} W

thus

⟨ℚU ⊗_{ℚP} Φ_{P} , ℚV ⊗_{ℚQ} Φ_{Q}⟩_{R} = ∑_{W ∈Irr_{ℚ}(R)} m_{W} n_{W} ⟨W, W ⟩_{R} .  (7.9)

Moreover

m_{W} ⟨W, W ⟩_{R} = ⟨W, ℚU ⊗_{ℚP} Φ_{P} ⟩_{R}
= ⟨ℚU^{op} ⊗_{ℚR} W, Φ_{P} ⟩_{P}
= m′_{W} ⟨Φ_{P} , Φ_{P} ⟩_{P} ,

and similarly n_{W} ⟨W, W ⟩_{R} = n′_{W} ⟨Φ_{Q}, Φ_{Q}⟩_{Q}. Hence

⟨ℚU ⊗_{ℚP} Φ_{P} , ℚV ⊗_{ℚQ} Φ_{Q}⟩_{R} = ∑_{W ∈Irr_{ℚ}(R)} m′_{W} n′_{W} \frac{⟨Φ_{P} , Φ_{P} ⟩_{P} ⟨Φ_{Q}, Φ_{Q}⟩_{Q}}{⟨W, W ⟩_{R}} .

(7.10)

Set σ = ⟨ℚU ⊗_{ℚP} Φ_{P} , ℚV ⊗_{ℚQ}Φ_{Q}⟩_{R}. Multiplying equations (7.9) and (7.10) shows that σ^{2} is equal to

( ∑_{W ∈Irr_{ℚ}(R)} m_{W} n_{W} ⟨W, W ⟩_{R})( ∑_{W ∈Irr_{ℚ}(R)} m′_{W} n′_{W} \frac{⟨Φ_{P} , Φ_{P} ⟩_{P} ⟨Φ_{Q}, Φ_{Q}⟩_{Q}}{⟨W, W ⟩_{R}} ) ,

hence

σ^{2} ≥ ∑_{W ∈Irr_{ℚ}(R)} m_{W} n_{W} m′_{W} n′_{W} ⟨Φ_{P} , Φ_{P} ⟩_{P} ⟨Φ_{Q}, Φ_{Q}⟩_{Q} .

Now if ⟨ℚU ⊗_{ℚP} Φ_{P} , ℚV ⊗_{ℚQ} Φ_{Q}⟩_{R} ≠ 0, then there exists W ∈ Irr_{ℚ}(R) such that the integers m_{W} and n_{W} are both positive, which happens if and only if m′_{W} and n′_{W} are both positive too. It follows that

σ^{2} ≥ ⟨Φ_{P} , Φ_{P} ⟩_{P} ⟨Φ_{Q}, Φ_{Q}⟩_{Q} .

Moreover equality can hold only if there is a unique W with this property, and in that case moreover m_{W} = m′_{W} = n_{W} = n′_{W} = 1. Thus the coefficient γ(W, kR_{ℚ}(U)(\overline{Φ}_{P})) of \overline{W} in the expression of kR_{ℚ}(U)(\overline{Φ}_{P}) as a linear combination of images of irreducibles (see Notation 4.1) is equal to \overline{m_{W}} = 1, and if W has type S, then \overline{Φ}_{S} ∈ H_{P} (S) by Lemma 4.3. Hence H_{S} ⊆ H_{P}. Similarly H_{S} ⊆ H_{Q}.

But also γ(Φ_{P} , kR_{ℚ}(U^{op})(\overline{W})) = \overline{m′_{W}} = 1, thus \overline{Φ}_{P} ∈ H_{S}(P), or equivalently H_{P} ⊆ H_{S}. Similarly H_{Q} ⊆ H_{S}, and finally H_{P} = H_{Q} = H_{S}. The last assertion of the lemma follows from Theorem 5.6.

Theorem 7.11 Let P be a finite p-group, and let V and V′ be irreducible ℚP -modules. If (T, S) is a genetic section of P for V, and (T′, S′) is a genetic section of P for V′, then

⟨(T, S) | (T′, S′)⟩_{ℤ} = \{ \begin{matrix}1 & if V ≅ V′ \\ 0 & otherwise \end{matrix} .

Proof : Let R (resp. R′) denote the factor group T/S (resp. T′/S′). The hypothesis of the theorem means that

V ≅ Ind_{T}^{P} Inf_{R}^{T}Φ_{R} ⟨V, V ⟩_{P} = ⟨Φ_{R}, Φ_{R}⟩_{R} ,

and similar conditions for V′ and R′. Let U (resp. U′) denote the set P/S (resp. the set P/S′), viewed as a P -set-R by left and right multiplication (resp. as a P -set-R′). Then V ≅ ℚU ⊗_{ℚR}Φ_{R}, and V′ ≅ ℚU′ ⊗_{ℚR′}Φ_{R′}. Hence

⟨V, V′⟩_{P} = ⟨ℚU ⊗_{ℚR} Φ_{R}, ℚU′ ⊗_{ℚR′} Φ_{R′}⟩_{P}
= ⟨Φ_{R}, ℚ(U^{op} ×_{P} U′) ⊗_{ℚR′} Φ_{R′}⟩_{R}

But U^{op} ×_{P} U′ is the set S\P/S′, viewed as an R-set-R′ (by left and right multiplication). Hence there is an isomorphism of R-sets-R′

U^{op} ×_{P} U′ ≅ ⨆_{x∈[T \P/T′]} S\T xT′/S′ ,

where [T \P/T′] is a set of representatives of double cosets. For each x ∈ [T \P/T′], the biset S\T xT′/S′ is transitive, and the stabilizer L_{x} of SxS′ in R × R′ is the set of pairs (rS, r′S′) such that rSxS′r^{′−1} = SxS′. Since rSxS′r^{′−1} = Srxr^{′−1}S′ for r ∈ T and r′ ∈ T′, this gives

L_{x} = {(rS, r′S′) ∈ R × R′ | r ∈ SxS′r′x^{−1}} .

The biset S\T xT′/S′ is isomorphic to (R × R′)/L_{x}, and this can be decomposed using notation and methods of sections 3.1 and 3.3 of [1] : the projection of L_{x} on R is equal to

p_{1,x} = p_{1}(L_{x}) = (SxT′x^{−1} ∩ T)/S = S(^{x}T′ ∩ T)/S ,

and the projection on R of L_{x} ∩ (R × 1) is equal to

k_{1,x} = k_{1}(L_{x}) = (SxS′x^{−1} ∩ T)/S = S(^{x}S′ ∩ T)/S .

Similarly

p_{2,x} = p_{2}(L_{x}) = S′(T′ ∩ T^{x})/S′ k_{2,x} = k_{2}(L_{x}) = S′(T′ ∩ S^{x})/S′ .

It follows that

q_{x} = p_{1}(L_{x})/k_{1}(L_{x}) ≅ (^{x}T′ ∩ T)/(^{x}S′ ∩ T)(^{x}T′ ∩ S)

and

q′_{x} = p_{2}(L_{x})/k_{2}(L_{x}) ≅ (T′ ∩ T^{x})/(S′ ∩ T^{x})(T′ ∩ S^{x}) .

The canonical isomorphism q_{x} → q′_{x} is the obvious isomorphism induced by conjugation by x. The group q_{x} is a factor group of the subgroup p_{1}(L_{x}) of R. It is isomorphic to R if and only if S(^{x}T′ ∩ T) = T and ^{x}S′ ∩ T ⊆ S. In particular q_{x} is isomorphic to both R and R′ if and only if (T, S) ^{x}(T′, S′).

These remarks show that ℚ(U^{op} ×_{P} U′) ⊗_{ℚR′} Φ_{R′} is isomorphic to

⨁_{x∈[T \P/T′]} Ind_{p_{1,x}}^{R}Inf_{q_{x}}^{p_{1,x}}Iso_{q′_{x}}^{q_{x}}Def_{q′_{x}}^{p_{2,x}}Res_{p_{2,x}}^{R′}Φ_{R′} ,

and this gives

⟨V, V′⟩_{P} = ∑_{x∈[T \P/T′]}⟨Def_{q_{x}}^{p_{1,x}}Res_{p_{1,x}}^{R}Φ_{R}, Iso_{q′_{x}}^{q_{x}}Def_{q′_{x}}^{p_{2,x}}Res_{p_{2,x}}^{R′}Φ_{R′}⟩_{q_{x}} . (7.12)

Moreover the sum in the right hand side is a sum of non-negative integers.

Let x ∈ [T \P/T′] such that (T, S) ^{x}(T′, S′). For such an element x one has that

p_{2,x} = R′ ,  q′_{x} = R′ ,  q_{x} = R ,  p_{1,x} = R .

It follows that Def_{q_{x}}^{p_{1,x}}Res_{p_{1,x}}^{R}Φ_{R} = Φ_{R}, and that Iso_{q′_{x}}^{q_{x}}Def_{q′_{x}}^{p_{2,x}}Res_{p_{2,x}}^{R′}Φ_{R′} is the image of Φ_{R′} by an isomorphism R → R′. Since Φ_{R} is the unique faithful rational irreducible representation of R, one has that

Iso_{q′_{x}}^{q_{x}}Def_{q′_{x}}^{p_{2,x}}Res_{p_{2,x}}^{R′}Φ_{R′} ≅ Φ_{R} .

It follows that

⟨V, V′⟩_{P} ≥ ⟨(T, S) | (T′, S′)⟩_{ℤ}⟨Φ_{R}, Φ_{R}⟩_{R} ≥ 0 .

Thus if V ≅ V̸′, one has that ⟨(T, S) | (T′, S′)⟩_{ℤ} = 0.

Conversely, suppose that V ≅ V′. Then

⟨V, V′⟩_{P} = ⟨V, V ⟩_{P} = ⟨Φ_{R}, Φ_{R}⟩_{R}

since R = T/S and (T, S) is a genetic section of P for V. Hence

⟨Φ_{R}, Φ_{R}⟩_{R} ≥ ⟨(T, S) | (T′, S′)⟩_{ℤ}⟨Φ_{R}, Φ_{R}⟩_{R} ≥ 0 .

Thus ⟨(T, S) | (T′, S′)⟩_{ℤ} ∈ {0, 1}.

Moreover R ≅ R′ by Corollary 5.9. Let S denote the set of elements x ∈ [T \P/T′] for which the corresponding term s_{x} in the right hand side of equation 7.12 is non-zero. By Lemma 7.8, if x ∈ S, then

s_{x}^{2} ≥ ⟨Φ_{R}, Φ_{R}⟩_{R}⟨Φ_{R′}, Φ_{R′}⟩_{R′} = ⟨Φ_{R}, Φ_{R}⟩_{R}^{2}

Hence s_{x} ≥ ⟨Φ_{R}, Φ_{R}⟩_{R}, and ∑_{x∈S} s_{x} = ⟨Φ_{R}, Φ_{R}⟩_{R}. Hence S consists of a single element x. Moreover by Lemma 7.8, the modules Def_{q_{x}}^{p_{1,x}}Res_{p_{1,x}}^{R}Φ_{R} and Iso_{q′_{x}}^{q_{x}}Def_{q′_{x}}^{p_{2,x}}Res_{p_{2,x}}^{R′}Φ_{R′} have a unique common irreducible direct summand W, which has type Σ such that H_{R} = H_{R′} = H_{Σ}.

If R (and R′) have order at least p^{2}, it follows from Theorem 5.6 that R ≅ R′ ≅ Σ. Moreover Σ is a section of q_{x}, which is a section of R. Thus q_{x} ≅ R ≅ R′ in this case, hence (T, S) ^{x}(T′, S′). Thus ⟨(T, S) | (T′, S′)⟩_{ℤ} is non-zero, hence equal to 1.

Now if R (and R′) are trivial, then V and V′ are isomorphic to the trivial module, thus (T, S) = (T′, S′) = (P, P), and

⟨(T, S) | (T′, S′)⟩_{ℤ} = 1 .

The remaining case is when R and R′ both have order p. In this case q_{x} is trivial or has order p. If it has order p, the conclusion is the same as before, as q_{x} ≅ R ≅ R′. And if q_{x} is trivial, then p_{1,x} is trivial or equal to R. It cannot be equal to R, for in that case Def_{q_{x}}^{p_{1,x}}Res_{p_{1,x}}^{R}Φ_{R} is a proper deflation of Φ_{R}, hence zero by Lemma 3.12. Thus p_{1,x} = 1, which means that T ∩^{x}T′ ⊆ S. Corollary 7.6 shows that S is a basic subgroup of P, and that T = S̃ = N_{P} (S). Now ^{x}T′ ∩ N_{P} (S) ≤ S, and the defining property of basic subgroups implies that |T′| ≤ |S|. But dim_{ℚ} V = |P : T |(p−1) = |P : T′|(p−1). Hence |T′| = |T | = p|S|. This contradiction shows that this case cannot occur, and this completes the proof of the theorem.

Remark 7.13 This theorem shows that if (T, S) and (T′, S′) are genetic sections for the same simple ℚP -module V, then there exists a unique double coset T xT′, for x ∈ P, such that (T, S) ^{x}(T′, S′). Such an element x defines an isomorphism

T/S →^{≅} (T ∩^{x}T′)/(S ∩^{x}S′) →^{≅} (T^{x} ∩ T′)/(S^{x} ∩ S′) →^{≅} T′/S′

which is well defined up to an inner automorphism of T/S (or T′/S′).

Theorem 7.14 Let P be a finite p-group. If L ∈ N_{1}, let S_{L} be the set of origins Q of P such that T_{Q}/S_{Q} ≅ L, and let [S_{L}] be a set of representatives of S_{L} modulo the relation ≑_{P}. Then

∀Q, R ∈ S_{L}, ⟨(T_{Q}, S_{Q}) | (T_{R}, S_{R})⟩_{ℤ} = \{ \begin{matrix}1 & if Q ≑_{P} R \\ 0 & otherwise \end{matrix} .

The images of the elements (T_{Q}, S_{Q}), for Q ∈ [S_{L}] form an orthonormal basis of S_{L,k}(P) = kB_{L}(P)/Rad⟨ | ⟩_{k}.

Proof : Let Q and R be origins in P. The corresponding simple modules V_{Q} and V_{R} are isomorphic if and only if Q ≑_{P} R. Moreover (T_{Q}, S_{Q}) is a genetic section of V_{Q}, by Proposition 7.4. Now the first assertion follows from Theorem 7.11. The images of the pairs (T_{Q}, S_{Q}), for Q ∈ [S_{L}], form an orthonormal set in kB_{L}(P). The cardinality of this set is equal to the number of equivalence classes of origins Q in P for which T_{Q}/S_{Q} ≅ L, modulo the relation ≑_{P}. This number is equal to the number of simple ℚP -modules of type L, i.e. the dimension of S_{L,k}(P) by Theorem 5.12, and this is equal to the rank of the form ⟨ | ⟩_{k}. This completes the proof of the theorem.

There is a similar result for the simple module S_{1,k} and the group B_{1}(P) :

Theorem 7.15 Let P be a finite p-group. Let S_{C_{p}} be the set of origins Q of P for which T_{Q}/S_{Q} ≅ C_{p}, and let [S_{C_{p}}] be a set of representatives of S_{C_{p}} modulo the relation ≑_{P}. The elements of B_{1}(P) defined by

u_{1} = (P, P) u_{Q} = (Q, Q) − (N_{P} (Q), N_{P} (Q)) for Q ∈ S_{C_{p}}

are such that

∀Q, R ∈ \{1\} ⊔ S_{C_{p}}, ⟨u_{Q} | u_{R}⟩_{ℤ} = \{ \begin{matrix}1 & if Q = R = 1 \\ p − 1 & if Q ≑_{P} R ≠ 1 \\ 0 & if Q ≑̸_{P} R \end{matrix} .

The images of the elements u_{X}, for X ∈ {1}⊔[S_{C_{p}}] form an orthogonal basis of S_{1,k}(P) = kB_{1}(P)/Rad⟨ | ⟩_{k}.

Proof : It is clear that

∀R ∈ \{1\} ⊔ S_{C_{p}}, ⟨u_{1} | u_{R}⟩_{ℤ} = \{ \begin{matrix}1 & if R = 1 \\ 0 & if R ≠ 1 \end{matrix} ,

because for any subgroups X and Y of P, one has that ⟨(X, X) | (Y, Y)⟩_{ℤ} = |X\P/Y |. In other words

⟨(X, X) | (Y, Y)⟩_{ℤ} = \overline{⟨ℚ(P/X), ℚ(P/Y)⟩_{P}} ,

where \overline{n} is the image in k of the integer n.

Now let Q and R be in S_{C_{p}}. Then

⟨u_{Q} | u_{R}⟩_{ℤ} = ⟨ℚ(P/Q) − ℚ(P/Q̃), ℚ(P/R) − ℚ(P/R̃)⟩_{P} = ⟨V_{Q}, V_{R}⟩_{P} .

This is zero if Q ≑̸_{P} R. Moreover if Q ≑_{P} R, then

⟨u_{Q} | u_{R}⟩_{ℤ} = ⟨V_{Q}, V_{R}⟩_{P} = ⟨Φ_{C_{p}}, Φ_{C_{p}}⟩_{C_{p}} = p − 1 .

Since p−1 = −1 in k, the images of the elements u_{X}, for X ∈ {1}⊔[S_{C_{p}}], form an orthogonal set of vectors of non-zero length in kB_{C_{p}}(P). The cardinality of this set is equal to 1 plus the number of equivalence classes of origins Q in P for which T_{Q}/S_{Q} ≅ C_{p}, modulo the relation ≑_{P}. This number is equal to the number of simple ℚP -modules of type 1 or C_{p}, i.e. the dimension of S_{1,k}(P) by Theorem 5.12, and this is equal to the rank of the form ⟨ | ⟩_{k}. This completes the proof of the theorem.

Remark 7.16 If Q is a basic subgroup of P, and if V_{Q} has type C_{p}, then Q is an origin. Indeed in this case, one has that

⟨V_{Q}, V_{Q}⟩_{P} = p − 1 = (p − 1)\frac{|N_{P} (Q) : Q|}{p}|I_{P} (Q, Q)| .

This implies that N_{P} (Q)/Q has order p, and that I_{P} (Q, Q) = N_{P} (Q). Thus T_{Q} = N_{P} (Q), and S_{Q} = Q. Now T_{Q}/S_{Q} is cyclic of order p, and Q is an origin.

References

[1] S. Bouc. Foncteurs d’ensembles munis d’une double action. J. of Algebra, 183(0238):664–736, 1996.

[2] S. Bouc. Tensor induction of relative syzygies. J. reine angew. Math., 523:113–171, 2000.

[3] S. Bouc. A remark on a theorem of Ritter and Segal. J. of Group Theory, 4:11–18, 2001.

[4] S. Bouc and J. Thévenaz. The group of endo-permutation modules. Invent. Math., 139:275–349, 2000.

[5] I. Bourizk. Certains foncteurs simples associés aux bi-ensembles. Thèse de l’Université Paris 7, 2001.

[6] W. Gorenstein. Finite groups. Chelsea, 1968.

[7] P. Roquette. Realisierung von Darstellungen endlicher nilpotente Gruppen. Arch. Math., 9:224–250, 1958.