n-Stabilizing Bisets

One generalises the notion of stabilizing bisets from [BouThe] to n-stabilizing bisets. This allows us to find new examples of stabilization for Roquette groups. We first investigate the idea of n-stabilizing bisets. We give a way to construct examples with the notion of idempotent bisets and n-expansive subgroups. Finally, for example, we look at Roquette groups and classify their n-stabilizing bisets.


Introduction
One purpose in representation theory is to try to describe representations of a finite group from a subgroup or subquotient of order as small as possible.This has been studied in Green's theory of vertices and sources and Harish-Chandra induction for reductive groups (see for instance Dipper & Du, 1997;Bouc, 1996).Another way to do so is to use stabilizing bisets introduced in Bouc and Thévenaz (2012).Indeed, let k be a field, G a finite group, U a (G, G)-biset and L a kG-module, where a (G, G)-biset U is a set which is both a left G-set and a right G-set such that (gu)h = g(uh), for all g ∈ G, h ∈ G and u ∈ U. Then U is said to stabilize L if U(L) := kU ⊗ kG L is isomorphic to L. If we suppose that L is indecomposable, then one can show that U is of the form Ind G A Inf A A/B Iso φ Def C C/D Res G C for some subgroups A, B, C, D and an isomorphism φ: C/D → A/B.In particular, this means that L can be obtained by a representation of A/B.Theorem 7.3 of Bouc and Thévenaz (2012) proves the existence of proper stabilizing bisets for simple modules, except when the group is Roquette and the module is faithful.Moreover, it seems impossible to find stabilizing bisets for the majority of Roquette groups.In order to obtain new examples, one generalizes this notion to n-stabilizing bisets, i.e. bisets U such that U(L) nL.This forces us to generalize the notions and results of Bouc and Thévenaz (2012).
It is shown in Bouc and Thévenaz (2012) that there is no stabilizing biset for Roquette p-groups.In this article, one shows that this is also true for Roquette groups with a cyclic Fitting subgroup.However, one finds non-trivial examples of n-stabilizing bisets for these groups.
We refer to Section 2 of Bouc and Thévenaz (2012), for the introduction to the notion of induction, inflation, deflation, restriction and isomorphism bisets and the corresponding notation.In particular, throughout this paper IndinfDefres stands for the biset Ind G A Inf A A/B Iso φ Def C C/D Res G C .We end this introduction with a short description of the organization of the paper, in Section 2 one finds some properties and characterizations of n-stabilization.In Section 3, one looks at n-stabilizing bisets and strong minimality.Then one looks at ways to obtain n-stabilizing bisets.We discuss one way with the help of n-idempotent bisets and characterize them completely.In Section 5 one generalises Section 6 of Bouc and Thévenaz (2012) by introducing the notion of n-expansive subgroups, this is another way to construct examples of n-stabilization.In this section, one also generalizes Section 3 of Bouc and Thévenaz (2012).
Finally, Section 6 is a study of examples.In particular, one treats Roquette p-groups, some simple groups and groups with a cyclic Fitting subgroup.One completely characterizes the n-stabilizing bisets for these examples.

n-Stabilizing Bisets
In this section one introduces the idea of n-stabilizing bisets.Using the notion of strongly minimality one could generalize Section 3 of Bouc and Thévenaz (2012).Theorem 12 is a generalization of Corollary 3.4 of Bouc and Thévenaz (2012) from the case of stabilization to that of n-stabilization.
Definition 1 (1) A section of a group G is a pair (A, B) of subgroups of G such that B is a normal subgroup of A.
(2) Two sections (A, B) and (C, D) of a group G are linked if We next quote Lemma 2.5 of Bouc and Thévenaz (2012): Proposition 2 (Generalized Mackey Formula) Let (A, B) and (C, D) be two sections of a finite group G. Then there is the following decomposition as a disjoint union of bisets is the butterfly biset and ψ is the composite Definition 3 Let U be a (G, G)-biset, let n be an integer and let L be a kG-module.Then U acts on L as follows U(L) is a kG-module and we say that U is applied to L. The biset U is said to n-stabilize L if U(L) nL.In the case n = 1, U is said to stabilize L.
Remark 4 We will focus our interest on indecomposable modules.If U = ∪ r i=1 U i is a decomposition of U as disjoint union of transitive bisets and if U n-stabilizes an indecomposable module L then Therefore by the Krull-Schmidt Theorem one has for every 1 ≤ i ≤ r that for some integer k i .For this reason, we shall assume that the biset U is transitive, hence, by Lemma 2.1 of Bouc and Thévenaz (2012), of the form Example 5 One refers to the last section of Bouc and Thévenaz (2012) for examples with n = 1.Here are examples with n > 1.Let k be an algebraically closed field of characteristic p and let P be a p-group.Let (A, B) be a section of P, where A and B are normal subgroups of P, and define L as Ind P A (k).By Green's indecomposability theorem L is indecomposable and then it's easy to see that U(L) = |P : A|L for U := Indinf P A/B Defres P A/B .Indeed, (A, B) = ( g A, g B) for all g in P because both A and B are normal, therefore using the Generalized Mackey Formula one has For example one can apply this to an extraspecial p-group P with B := Z(P) and A := N P ( x ), where x is a non-central element of order p; or also to P the dihedral group D 8 of order 8 with A = r and B = r 2 , where r is the rotation by an angle of π/2.(2) The biset U is said to be strongly minimal if, for any transitive biset Proof.Suppose U is not strongly minimal.Let is isomorphic to copies of the trivial module k and thus nL = ν Ind G A (k) for some integer ν ≥ 1.But the trivial kG-module is always a submodule of Ind G A (k), which contradicts the assumption that L is not the trivial module.Therefore such U cannot exist and U is strongly minimal.
Theorem 9 Consider two transitive (G, G)-bisets Let L be an indecomposable kG-module such that U(L) nL and U (L) mL for n, m ∈ N. Let M = Defres G C/D (L) and suppose U is strongly minimal.Let g be an element of G. Then only two cases are possible: , where β(g) is the isomorphism corresponding to the linking between the sections ( g A, g B) and Proof.Applying successively U and U one obtains Therefore, by the Krull-Schmidt theorem, one has, for all g ∈ [C \G/A], In other words, one has a k g -stabilizing biset for L, for a certain k g ∈ N. If k g 0 and because U is strongly minimal, the biset Btf(C , D , g A, g B) must be reduced to Indinf C /D (C ∩ g A)D /(C ∩ g B)D Iso β(g) , where β(g) is the isomorphism corresponding to the linking between the sections ( g A, g B) and ((C ∩ g A)D , (C ∩ g B)D ).Indeed, otherwise Btf(C , D , g A, g B) would go through a subsection of (A, B), which is a contradiction to the fact that U is strongly minimal.If k g = 0, then the module Btf(C , D , g A, g B) Conj g Iso φ (M) is zero, as the operation Indinf G A /B Iso φ cannot annihilate a module.For such g, the section ( g A, g B) is not linked to ((C ∩ g A)D , (C ∩ g B)D ) as otherwise the biset Btf(C , D , g A, g B) would have been reduced to Iso β(g) , but the latter does not annihilate Conj g Iso φ (M).
Remark 10 Let M be the module Defres G C /D (L).Using the same notation, we observe that one has Corollary 11 Using the same notation and hypotheses as in Theorem 9 and suppose that both U and U are strongly minimal.Let g be an element of G.
(1) Only two cases are possible: (i) The module Btf(C , D , g A, g B) Conj g Iso φ (M) is zero and the section (ii) The biset Btf(C, D, g A, g B) is reduced to Iso β(g) , where β(g) is the isomorphism corresponding to the linking between the sections ( g A, g B) and (C , D ).
Let M be the set of elements of [C \G/A] such that we are in case (ii) and let d be the cardinality of M .
(3) One has the following equality nm = dd , where d is the number of double cosets ChA such that Proof.One uses the same argument as in the proof of Theorem 9 but suppose now that U is strongly minimal.One deduces that Btf(C , D , g A, g B) is reduced to an isomorphism if k g 0, because U and U are strongly minimal.This means that, if k g 0, Indinf G A /B Iso φ Iso β(g) Conj g Iso φ (M) k g L. In particular if k g 0, the dimension on the right hand side does not depend on g, because on the left of the isomorphism it does not.Therefore all non-zero k g are equal.The isomorphism becomes By looking at the dimension in this equality, one obtains that where d is the number of double cosets C gA such that k g 0.
Exchanging the roles of U and U in the previous argument one has mn = k h d , where d is the number of double cosets ChA such that k h 0 and Furthermore, using Remark 10, one has By looking at the dimension one obtains that n dim M = d dim M. Exchanging the roles of U and U in the previous argument one has m dim M = d dim M .Finally, using these two equations, one obtains that mn = dd and that k g = d and k h = d, whenever k g and k h are non-zero.
Theorem 12 Let U = Indinf G A/B Iso φ Defres G C/D be a strongly minimal n-stabilizing biset for an indecomposable kG-module L. Let M = Defres G C/D (L).Then, there exist n double cosets CgA such that (1) Btf(C, D, g A, g B) Conj g Iso φ (M) {0}, (2) the sections (C, D) and ( g A, g B) are linked, (3) the module M is invariant under β(g)c g φ, where β(g) is the isomorphism corresponding to the linking between the sections (C, D) and ( g A, g B), ( 4) if h ∈ G does not belong to one of these cosets, the section ( h A, h B) is not linked to (C, D).
Proof.Using part 3 of Corollary 11 with U = U, m = n and d = d, one obtains that n = d.Therefore by definition of d, there exist exactly n double cosets CgA such that Btf(C, D, g A, g B) Conj g Iso φ (M) {0}.For these double cosets one knows that Btf(C, D, g A, g B) is reduced to Iso β(g) , where β(g) is the isomorphism corresponding to the linking between the sections ( g A, g B) and (C, D).In particular, the sections (C, D) and ( g A, g B) are linked.If h ∈ G does not belong to one of these cosets, the section ( h A, h B) cannot be linked to (C, D), otherwise we would have another non-zero module of the form Btf(C, D, h A, h B) Conj h Iso φ (M).
Finally one proves (3).By the Krull-Schmidt Theorem we can write M as where the M jr j 's are indecomposable and pairwise non-isomorphic, f ( j) is an integer depending on j and a j < a j+1 for all j.Using the second part of Corollary 11 and the fact that Note that M 11 appears in the decomposition of Iso β(g i )c g i φ (M) for all i = 1, . . ., n.Indeed, Iso β(g i )c g i φ sends an indecomposable module to an indecomposable module and if Iso β(g i )c g i φ (M j 1 r j 1 ) Iso β(g i )c g i φ (M j 2 r j 2 ) then M j 1 r j 1 M j 2 r j 2 by applying Iso (β(g i )c g i φ) −1 on both sides.As the M jr j are all pairwise non-isomorphic this means that there is the same number of indecomposable modules in M than in Iso β(g i )c g i φ (M) and that the indecomposable modules in the decomposition are the same.Denote by m i the multiplicity of M 11 in Iso β(g i )c g i φ (M), then m i ≥ a 1 for all i = 1, . . ., n, as for all i the module M 11 corresponds to Iso β(g i )c g i φ (M j i r i ) for some M j i r i , which means a j i ≥ a 1 for all i.Moreover, looking at the two decompositions of nM one has n i=1 m i = na 1 and so m i = a 1 for all i.Applying this argument to all the modules M 1r 1 one obtains that, for all i, Using this result, the same argument proves that Finally, continuing like this, one has, for all i The next three results are generalized forms of respectively Corollary 3.5, Proposition 4.3 and Proposition 8.5 of Bouc and Thévenaz (2012).We omit the proofs here as they are similar to the case n = 1.We refer to Monnard (2014) for the proofs.
Corollary 13 Let U = Indinf G A/B Iso φ Defres G C/D be a strongly minimal n-stabilizing biset for an indecomposable kG-module L. Then there exists a section ( Ã, B) linked to (C, D) by σ such that L is n-stabilized by Proposition 17 Let G be a group and L a faithful kG-module such that L is n-stabilized by As nL is faithful, the latter is trivial and so too is the G-core of B. Proposition 18 Let G be a group and L a faithful simple kG-module such that L is n-stabilized by and the result follows.

n-Stabilizing Bisets and Strong Minimality
In this section one treats the question of strong minimality and existence of strongly minimal n-stabilizing bisets.
Proposition 20 Let G be a finite group, U be a n U -stabilizing biset of the form Indinf G A/B V Iso φ Defres G C/D for a kG-module L and V a strongly minimal n V -stabilizing biset for M := Iso φ Defres G C/D (L).Moreover suppose that M is indecomposable.Then U is strongly minimal.
Proof.Set V = Indinf A/B H/J Iso σ Defres A/B S /T and let W be a n W -stabilizing biset for L. Set W = Indinf G A /B Iso φ Defres G C /D .We have to show that |H/J| ≤ |A /B |.Using these settings, one has Using the Generalized Mackey Formula, the left hand side becomes where the sum is taken over g ∈ [C\G/A ] and h ∈ [C \G/H].Because M is indecomposable, this implies that for each summand there exists a certain k g,h such that Note that k g,h 0 for at least one pair (g, h).The biset V is strongly minimal therefore the biset Btf(C , D , h H, h J) has to be reduced to Proposition 21 Let G be a finite group and let U := Indinf G A/B V Iso φ Defres G C/D be a strongly minimal n U -stabilizing biset for an indecomposable kG-module L, where V n V -stabilizes M := Iso φ Defres G C/D (L).Then V is strongly minimal.

Proof.
Set V = Indinf A/B H/J Iso σ Defres A/B S /T and let W be a n W -stabilizing biset for M.
Using Mackey's Formula, the first term on the left becomes Because L is indecomposable, this implies that for each summand there exists a certain k g such that and k g 0 for at least one g.By strongly minimality of U the biset Btf(S , T, g H , g J ) must, at least, be reduced to Iso ψ Defres g H / g J (S ∩ g H ) g J /(T ∩ g H ) g J , which means that (S , T ) is linked to a subsection of ( g H , g J ).In particular |H/J| = |S /T | ≤ |H /J |, which proves the strongly minimality of V.
Proposition 22 Let G be a finite group, U := Indinf G A/B Iso φ Defres G C/D and L a kG-module n U -stabilized by U. Suppose M := Iso φ Defres G C/D (L) is indecomposable.Then there exists a biset V, n V -stabilizing M, such that W := Indinf G A/B V Iso φ Defres G C/D is strongly minimal for L. Moreover V is strongly minimal for M. Proof.One proves this by induction hypothesis to |G|.If |G| = 1, then the trivial biset is strongly minimal.Now suppose the statement is true for groups of order less than |G|.If U is strongly minimal then V = Id.Suppose U is not strongly minimal.Moreover suppose |A/B| < |G| and apply the induction on the indecomposable module M with the identity as stabilizing biset.So one obtains a strongly minimal biset V such that V(M) n V M. By Proposition 20 the biset is strongly minimal for L.
It is left to consider the case |A/B| = |G|.This implies that U = Iso φ , but U is not strongly minimal by assumption, therefore there exists a proper biset V 1 , i.e. not reduced to an isomorphism, such that the argument of the first case, one obtains a strongly minimal n V -stabilizing biset V for the module L and therefore W = V Iso φ is strongly minimal for L.
Remark 23 Note that W is a n U n V -stabilizing biset for L and not simply a n U -stabilizing biset.
Proposition 24 Let G be a finite group, U := Indinf G A/B Iso φ Defres G C/D and L an indecomposable kG-module stabilized by U. Then there exists a biset V such that U := Indinf G A/B V Iso φ Defres G C/D is minimal for L. Moreover V is minimal for M := Iso φ Defres G C/D (L).Proof.Following exactly the proof of Proposition 22 with n U = 1, the fact that M is indecomposable because Indinf G A/B (M) L is and the notion of minimality instead of strongly minimality, one obtains the result.Proposition 25 Let L be a faithful simple kG-module.Suppose that whenever U(L) L for U a minimal biset then U is reduced to an isomorphism.Then, for an arbitrary biset Proof.By proposition 24 there exist subgroups H and J with J a normal subgroup of H with B ≤ H ≤ A and is minimal for L. As a minimal stabilizing biset one has, by hypothesis, that J = {1} and H = G and so in particular B = {1} and A = G.
One gives here a complete classification of such bisets.

if and only if the following three conditions hold:
1) There are n (C, A)-double cosets.
2) The sections (C, D) and ( g A, g B) are linked for all g.
3) For every g ∈ G, there exist x ∈ N G ( g A, g B) and y ∈ N G (C, D) such that where β(g): C/D → g A/ g B is the isomorphism induced by the linking.
Proof.The idea of the proof is similar to that of Theorem 5.1 of Bouc and Thévenaz (2012).We refer to Theorem 2.26 of Monnard (2014) for more details.
As Proposition 5.4 of Bouc and Thévenaz (2012), one obtains the following generalized result: Proposition 28 Let U be an n-idempotent (G, G)-biset.For any kG-module L , the kG-module L := U(L ) is n-stabilized by U.
Remark 29 Note that in general L need not be indecomposable.
(2) If A and B are normal subgroups of G and U := Indinf G A/B Defres G A/B , then U is |G : A|-idempotent.Indeed, one has |G : A| (A, A)-double cosets.By normality the sections are trivially linked and by taking x = y = 1 the third condition is also fulfilled.This is the case, in particular, of Example 5.

As this occurs exactly n times, one concludes that Defres
The second claim in this theorem follows from the first and the definition of L.
Example 35 Here is an example of n-expansivity in S 6 .
The second one satisfies the second part of Definition 31 and the two others the first part.Therefore T is another example of a 2-expansive subgroup in S 6 .Again, setting M to be the sign representation of S /T one obtains an example of a 2-stabilizing biset, but the module L := Indinf S 6 S /T (M) is not indecomposable over C.

n-Stabilizing Bisets and Roquette Groups
In Bouc and Thévenaz (2012), Theorem 7.3 states that if k is a field, G a finite group and L a simple kG-module, then there exists an expansive subgroup T of G such that This theorem proves the existence of stabilizing bisets for simple modules.However, it is possible that this biset is trivial, i.e. it is reduced to an isomorphism.The proof of the theorem shows that this could only be the case if G is Roquette and L is faithful.Recall that a finite group G is said to be a Roquette group if all its normal abelian subgroups are cyclic.
This raises the question of proving the existence, or non-existence, of stabilizing bisets for Roquette groups and more generally of n-stabilizing bisets.The goal of this section is to study n-stabilization for Roquette groups.Let G be a Roquette group and denote by F(G) the Fitting subgroup of G, which is the product of the normal subgroups O p (G) for all primes p.As G is Roquette each O p (G) does not contain a characteristic abelian subgroup that is not cyclic.By Theorem 4.9 of Gorenstein (1980), such groups are known.More precisely, each subgroup O p (G) is the central product of an extraspecial group with a Roquette p-group.Roquette p-groups are known, see Chapter 5, Section 4 of Gorenstein (1980), so one starts our study with these groups.Then, one continues with groups with a cyclic Fitting subgroup, corresponding to cyclic O p (G) for every prime p.

Roquette p-Groups
The case of Roquette p-groups has already been studied in Bouc and Thévenaz (2012).It is shown that if U is a stabilizing biset for a faithful simple module, then U has to be reduced to an isomorphism, see Theorem 9.3.One will discuss the case of n-stabilizing bisets for n > 1.
Theorem 36 Let p be a prime number and let P be a Roquette p-group of order p k+1 .Let U := Indinf G A/B Iso φ Defres G C/D be a n-stabilizing biset for L where L is a simple faithful CP-module.Then one has B = D = 1.
Proof.First note that by 17 and 18, the P-cores of B and D are trivial.In particular, B ∩ Z(P) and D ∩ Z(P) have to be trivial, as these intersections are contained in the P-core of, respectively, B and D. It follows from Lemma 9.1 of Bouc and Thévenaz (2012) that B and D are trivial, except possibly if p = 2, P is dihedral or semi-dihedral, and B and D are non-central subgroups of order 2. Therefore one has four cases to treat • B and D are non-central subgroups of order 2, • B is a non-central subgroup of order 2 and D = 1, • B = 1 and D is a non-central subgroup of order 2, • B = 1 and D = 1.
One starts with a general remark on the first three cases that occur only if P is dihedral (with k ≥ 3), or semidihedral (with k ≥ 3).As L is a simple faithful module, by looking at the character tables of D 2 k+1 and S D 2 k+1 , one sees that the dimension of L is 2. Also the character of Res P C 2 ×Z(P) (L), for C 2 a non-central subgroup of order 2, is the following where c generates C 2 and z generates Z(P).Thus the module χ Res P C 2 ×Z(P) (L) splits in the sum of the following two characters of degree one Therefore, Defres P C 2 ×Z(P)/C 2 (L) is the sign representation.One proves now that the first three cases are impossible.Consider first the case where B is a non-central subgroup of order 2 without assumption on D. By Lemma 9.1 of Bouc and Thévenaz (2012), one knows that N P (B) = B× Z(P).This fact forces us to have A = N P (B), otherwise the A/B-module M = Iso φ Defres P C/D (L) would be trivial and by Proposition 15 the module L would be trivial as well, but this contradicts the fact that L is faithful.As A/B is of order 2, the module M is therefore forced to be copies of the sign representation M 1 .As L is of dimension 2, either M = M 1 or M = 2M 1 .We would like to know if Ind P A (Inf A A/B (M)) is a sum of copies of L. To do so one uses the scalar product on characters and Frobenius reciprocity The latter equality holds because, as described in the general remarks above, Res P A (L) is the sum of two nonisomorphic represention of degree 1.It is easy to check that one of them is Inf A A/B (M 1 ).Thus at most two copies of L are in the decomposition of Ind ) contains other modules, non-isomorphic to L, in its decomposition which implies that it cannot be the sum of n copies of L.
Assume now that B = 1 and D is a non-central subgroup of order 2. As above one has C = N P (D) = D × Z(P) and M is the sign representation.Moreover the subgroup A is of order 2 as A is isomorphic to C/D.We would like to know if Ind P A (M) is a sum of copies of L. Again using the scalar product one has The latter inequality occurs because L is of dimension 2 and therefore the sign representation can only occur twice.
In fact, it is easy to see that it is equal to 2 if A = Z(P) and 1 otherwise.In any case one has This means again that Ind P A (M) contains other modules, non-isomorphic to L, in its decomposition and so it cannot be the sum of n copies of L.
Finally we are restricted to the last case, namely B = {1} = D and the result follows.
Decompose Ind H A (M) as the sum of irreducible H-modules V i and using the remark above on the induction on modules from a maximal subgroup, one obtains that with, for all i, L i1 and L i2 two non-isomorphic irreducible P-modules.Thus the module Ind P A (M) cannot be only n copies of a module L.

Groups With a Cyclic Fitting Subgroup
In this section one proves that if G is a solvable group such that F(G) = C n = i C p k i i and U is a stabilizing biset for a simple faithful CG-module, then U has to be reduced to an isomorphism.Then one describes the case of ν-stabilizing bisets as one did for Roquette p-groups, where ν is an integer.In this section, G is assumed to be solvable.Suppose n = 2 k p k 1 1 . . .p k m m for some distinct odd primes p i and integers k and k i , so First note that it is a well known fact that C G (F(G)) ≤ F(G) and therefore G/F(G) injects into Out(F(G)).Thus one has the following exact sequence where S is a subgroup of Aut(C n ).The map ι: C n → G is the inclusion map.The map π: G → S sends an element g to the conjugation map c g .Suppose moreover that S is a subgroup of C 2 × i C p i −1 where C 2 is either generated by This added condition is to ensure that G is Roquette, see Theorem 3.7 of Monnard (2014).We start with a number of general lemmas.
Lemma 38 Let G be an extension of S by C n as above.Let D be a subgroup of G such that D But CC 2 k decomposes as the sum of all simple CC 2 k -modules.Using the Krull-Schmidt Theorem and the fact that Ind Proof.The idea of this proof is to restrict the module L to certain well-chosen subgroups using once Clifford's Theorem and then Mackey's Formula as νL can be written as U(L).Then one utilizes the fact that these two decompositions should be isomorphic. where The kernel of N is Q but, as mentioned before, Res G H (L) is a sum of faithful modules, therefore Q is trivial and so H ∩ A = H.This in turn implies that H ≤ A and therefore normalizes B, because B is normal in A. This implies that B acts trivially on H by Lemma 38.Therefore B is either trivial or π(B) is generated by β 1 or β 2 , where π denotes the homomorphism from G to S .Suppose the latter holds, so k > 2. By Clifford's Theorem where L 1 is a simple C 2 k -module and it is a subgroup of I 1 /C n and so the order of G/I 1 is at most 2.This implies that there are at most 2 non-isomorphic modules appearing in Res G C 2 k (L).Next we note that where the last equality holds because either for β 1 or β 2 one has Using this remark and Mackey's Formula we restrict L to C 2 k :

Now Res A
C 2 ( M) decomposes as a sum of representations that are either the trivial or the sign representation, but the trivial cannot occur.Indeed suppose the trivial representation T + appears in the decomposition of Res A C 2 ( M).Then Ind C 2 k C 2 (T + ) is not a faithful representation as C 2 is in its kernel, contrary to the fact that Res G C 2 k (L) is faithful.Therefore Res A C 2 ( M) is a sum of copies of the sign representation T − and Ind C 2 (T − ) decomposes as the sum of all faithful representations of C 2 k by Lemma 39 and there are 2 k−1 such non-isomorphic representations.So the module Res G C 2 k (L) decomposes with at least 2 k−1 non-isomorphic representations.As k > 2 one has 2 k−1 > 2 and so a contradiction is obtained with the decomposition using Clifford's Theorem.Therefore the only possibility is that B = {1}.Thus our purpose is to show that Ind G F(G) (ξ ⊗ Ir j ) is not isomorphic to L = Ind G F(G) (ξ) for at least one representation Ir j .To do so, one proves that ξ ⊗ Ir is not conjugate, by an element of G/F(G) to ξ, where Ir denotes a non-trivial C[F(G)/(F(G) ∩ A)]-module.We specify which Ir is taken later on.
Let p be a prime dividing |F(G) : A ∩ F(G)| and let i be its highest power dividing |F(G) : A ∩ F(G)|.Choose p such that p i is strictly smaller that p k , where k is the highest power of p such that p k divides n.As F(G) is cyclic, one decomposes Ir as the tensor product of a representation θ of C p i and a representation θ c of its complement in F(G)/(F(G) ∩ A), i.e.Ir = θ ⊗ θ c .Note that θ is a p i th root of unity.In the same fashion ξ = ξ 1 ⊗ ξ 2 , where ξ 1 is a p k th root of unity and ξ 2 is a representation for C n/p k .Then one has One now sets Ir such that θ = ξ p k−i 1 and then one has ξ 1 ⊗ θ = ξ 1+p k−i 1 .Because of the assumption on S made at the beginning of the section, this representation cannot be conjugate to the representation ξ 1 by an element of G/F(G).Indeed, such an element would have order a divisor of p i , as such an element must be of the following form Moreover, it is easy to check that α δ (ξ 1 ) = ξ 1+δp k−i 1 and so α p i = id.So ξ ⊗ Ir is not conjugate to ξ.Finally, one has proved that Ind G F(G) (ξ ⊗ Ir) L = Ind G F(G) (ξ) and therefore other modules than L appear in the decomposition of U(L).

Proposition 6
Let U := Indinf G A/B Iso φ Defres G C/D be an n-stabilizing biset for a module L. Let M := Defres G C/D (L).Then n = |G:A|dimM dimL .In particular, one has n ≤ |G|.Proof.By taking the dimension of U(L) nL, one has ndimL = |G : A|dimDefres G C/D (L).Therefore one has n = |G:A|dimM dimL .As dimM ≤ dimL, one has n ≤ |G : A| ≤ |G|.Definition 7 Let U = Indinf G A/B Iso φ Defres G C/D be a biset n-stabilizing a kG-module L. (1) The biset U is said to be minimal if, for any transitive biset U = Indinf G A /B Iso φ Defres G C /D n-stabilizing L, we have |C/D| ≤ |C /D |.
is a quotient of L and N acts trivially on it; however, since L is simple and faithful one must have N = {1}.Proposition 19 Let k be a field and let U = Indinf G A/B Iso φ Defres G C/D be a biset n-stabilizing a simple kG-module L. Then n|A| ≥ |N G (D)| and in particular n|A| ≥ |C|.Proof.By the proof of Proposition 8.1 of Bouc and Thévenaz (2012), one has dim L ≤ |G : N G (D)| dim Defres G N G (D)/D (L).By Lemma 6, one has n dim L = |G : A| dim Defres G C/D (L).Moreover, dim Defres G N G (D)/D (L) is equal to dim Defres G C/D (L) as it only depends on the action of D on L. Therefore |G linked to a subsection of (C , D ).In particular |H/J| ≤ |C /D | = |A /B |, which proves the strong minimality of U.

Proof.
For the first equality, let x ∈ N C n (D).Then, for all d ∈ D one hasxdx −1 ∈ D. But xdx −1 = x d x −1 d which belongs to D if and only if x d x −1 = 1 that is x = d x .This implies that x is an element of C C n (D).The other inclusion is trivial.For the second equality, note that the action of D on C n is the same as the action of π(D) on C n by definition of the map π.Lemma 39 Let C 2 k be a cyclic group of order 2 k and C 2 its subgroup of order 2. Denote by T + and T − the trivial and the sign C-representation of dimension 1 of C 2 .Then the module Ind C 2 k C 2 (T + ) decomposes as the sum of all nonfaithful representations of C 2 k and the module Ind C 2 k C 2 (T − ) decomposes as the sum of all faithful representations of C decomposes as the sum of all non-faithful representations of C 2 k .Therefore the module Ind C 2 k C 2 (T − ) has to decompose as the sum of all faithful representations of C 2 k .Theorem 40 Let G be a Roquette group with F(G) = C n .Let U := Indinf G A/B Iso φ Defres G C/D be a ν-stabilizing biset for L, where L is a simple faithful CG-module.Then B = {1} and A contains C 2 C p 1 . . .C p m .
Theorem 41 Let G be a Roquette group with F(G) = C n .Let U := Indinf G A/B Iso φ Defres G C/D be a stabilizing biset for L, where L is a simple faithful CG-module.Then one has (A, B) = (C, D) = (G, 1).Proof.By Proposition 25 it is sufficient to look at minimal stabilizing bisets.If U is minimal, one knows that if B = {1} then A = G by Proposition 8.4 ofBouc and Thévenaz (2012), but Theorem 40 shows that B = {1} and so the results follows.Ind G A∩F(G) Res F(G) A∩F(G) ( g ξ) |A\G/F(G)| Ind G A∩F(G) Res F(G) A∩F(G) (ξ) |A\G/F(G)| Ind G F(G) Ind F(G) A∩F(G) Res F(G) A∩F(G) (ξ).Using Frobenius reciprocity one has IndF(G) A∩F(G) Res F(G) A∩F(G) (ξ) ⊕ j ξ ⊗Ir j where {Ir j } is a set of isomorphism classes of simple C[F(G)/(F(G)∩A)]-modules.The sum is not reduced to one module as A∩F(G) F(G) by assumption.This means that U(L) j |A\G/F(G)| Ind G F(G) (ξ ⊗ Ir j ).
By Proposition 17, one knows that B has a trivial G-core.Therefore B ∩ C n = {1}.Denote by M the A-module Inf A A/B Iso φ Defres G C/D (L) and by H the product C 2 C p 1 . . .C p m .Using Clifford's Theorem one has V) = g ker(V) = ker(V), as the subgroups of H are characteristic.Now by Mackey's Formula Let Q be a complement of H ∩ A in H.Such a complement exists because H ∩ A ≤ C 2 C p 1 . . .C p m and so Q = C |H|/|H∩A| .Now one extends Res A H∩A ( M) to an H-module N by saying that Q acts trivially on N.
H (νL) ν Res G H (L) ν ⊕ g∈G/I μ g Vwhere V is a simple H-module and I := {g ∈ G | g V V}.As L is faithful the module Res G H (L) is also faithful and so is V, because ker( g