biset functors and genetic sections for p-groups

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Journal of Algebra 284 (2005) 179–202

www.elsevier.com/locate/jalgebr

Biset functors and genetic sections forp-groups

Serge Bouc

UFR de Mathématiques, Université Paris7–Denis Diderot, 75251 Paris cedex 05, France

Received 25 May 2004

Available online 11 September 2004

Communicated by Michel Broué

Abstract

In this note I show that ifF is a biset functor defined over finitep-groups, then for each finitp-groupP , there is a direct summand ofF(P ) admitting a natural direct sum decomposition indexby the irreducible rational representations ofP , or equivalently, by the equivalence classes of origin P , or also by equivalence classes of genetic sections ofP . This leads to a description of the torsiopart of the group of relative syzygies in the Dade group ofP , and to a conjecture on the structurethe torsion part of the whole Dade group ofP . 2004 Elsevier Inc. All rights reserved.

1. Basic subgroups and origins

1.1. In Sections 1–3 of this paper I recall some notation and definitions which arein the statement of the main theorem (Theorem 3.2). Section 4 is devoted to some proof origins and genetic sections. Section 5 is the proof of Theorem 3.2, and Sectionapplication to the Dade group. In Section 7, I use the notion ofrational biset functor togive a description of the torsion part of the functor of relative syzygies in the Dade g

E-mail address:[email protected].

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.06.034

180 S. Bouc / Journal of Algebra 284 (2005) 179–202

1.2. Throughout this paper, the letterp denotes a fixed prime number. IfP is a finitep-group, denote byRQ(P ) the Grothendieck group of finitely generatedQP -modules.There is a natural bilinear form onRQ(P ), with values inZ, defined by

〈V,W 〉P = dimQ HomQP (V,W) for QP -modulesV andW.

Recall some definitions from [1].

1.3. Definition [1, 2.3]. Let P be a finitep-group. A subgroupQ of P is called basic if thefollowing two conditions hold:

(1) the quotientNP (Q)/Q is cyclic or generalized quaternion;(2) if R is any subgroup ofP such thatR ∩ NP (Q) ⊆ Q, then|R| |Q|.

1.4. Associated simple modules

If Q is a proper basic subgroup ofP , then there is a unique subgroupQ ⊃ Q of P with|Q : Q| = p, and the kernel of the projection map

QP/Q → QP/Q

is an irreducibleQP -module, denoted byVQ.The groupP itself is a basic subgroup ofP , and by conventionVP is the trivialQP -

moduleQ. With this notation, any irreducibleQP -module is isomorphic toVQ, for somebasic subgroupQ of P .

If Q andQ′ are basic subgroups ofP , then theQP -modulesVQ andVQ′ are isomorphicif and only if Q P Q′, whereP is the relation defined in [1, 2.7] by

Q P Q′ ⇔ |Q| = |Q′| and ∃x ∈ P, Qx ∩ NP (Q′) ⊆ Q′.

1.5. Notation and definition [1, 2.4]. ForQ andR subgroups ofP , set

IP (Q,R) = x ∈ P | Qx ∩ NP (R) ⊆ R

, IP (Q,R) = Q\IP (Q,R)/NP (R).

If V is a simpleQP -module, andW is a finitely generatedQP -module, denote bym(V,W) the multiplicity ofV as a direct summand ofW .

If Q is a basic subgroup ofP andR is any subgroup ofP , then (see [1, 2.5])

m(VQ,QP/R) = |IP (R,Q)||IP (Q,Q)| .

In particular, if this is non-zero, then|R| |Q|.

S. Bouc / Journal of Algebra 284 (2005) 179–202 181

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1.6. Definition [1, 7.3]. If Q is a basic subgroup ofP , set

TQ = ⟨IP (Q,Q)

⟩, SQ =

⋂x∈TQ

xQ.

The basic subgroupQ of P is called an origin (inP ) if TQ/SQ has normalp-rank 1, i.e.if it is cyclic, generalized quaternion, semi-dihedral, or dihedral of order at least 16.

If V is a simpleQP -module, there exists an originQ in P such thatV ∼= VQ (see [1,7.5]). ThenQ is called an origin ofV , andV is said to havetypeTQ/SQ. One can showthat the type ofV is well defined up to isomorphism (see [1, 7.5]). IfTQ/SQ is cyclic orgeneralized quaternion, thenSQ = Q andTQ = NP (Q) (see [1, 7.6]). IfTQ/SQ is dihedralor semi-dihedral, then|Q : SQ| = 2 (see [1, 7.4]).

If R is a finitep-group of normalp-rank 1, then there is a unique faithful irreducibQR-moduleΦR , up to isomorphism (see [1, 3.7]).

1.7. Definition. A pair (T ,S) of subgroups ofP with S T is called a section ofP . Twosections(T ,S) and(T ′, S′) are said to be linked (notation(T ,S) (T ′, S′)) if

T ∩ S′ = S ∩ T ′ and T .S′ = S.T ′.

They are linked moduloP (notation(T ,S) P (T ′, S′)) if there existsx ∈ P such that(T ,S) (xT ′, xS ′

).The section(T ,S) is called genetic ifT/S has normalp-rank 1, and if setting

V = IndPT InfTT/S ΦT/S,

there is aQ-algebra isomorphism

EndQP (V ) ∼= EndQT/S(ΦT/S).

In particular, it follows that EndQP (V ) is a (skew) field, henceV is an irreducibleQP -module, called the irreducibleQP -moduleassociatedto (T ,S). The pair(T ,S) is calleda genetic section ofP for V , or associated toV . If Q is an origin inP , then(TQ,SQ) is agenetic section ofP associated toVQ (see [1, 7.4]).

The section(P,P ) of P is called thetrivial genetic sectionof P . It is the unique genetisection(T ,S) of P for whichT = S: indeedΦ1 = Q, and IndPT InfT1 Φ1 = QP/T is simpleif and only if T = P , sinceQ is always a direct summand ofQP/T .

If (T ,S) and (T ′, S′) are genetic sections ofP , and if V and V ′ are the associatesimple QP -modules, thenV ∼= V ′ if and only if (T ,S) P (T ′, S′) (see [1, 7.11]). Inparticular, the relation P is an equivalence relation on the set of genetic sections oP .Recall moreover that if(T ,S) P (T ′, S′), then there exists a unique double cosetT xT ′in P such that(T ,S) x(T ′, S′).


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2. Biset functors

2.1. Notation and definition. Let k be a commutative ring. Denote byCp,k the followingcategory:

• the objects ofCp,k are the finitep-groups;• if P and Q are finitep-groups, then HomCp,k

(P,Q) = k ⊗Z B(Q × P op), whereB(Q × P op) is the Burnside group of finite(Q,P )-bisets;

• the composition of morphisms isk-bilinear, and ifP , Q, R are finitep-groups, ifUis a finite (Q,P )-biset, andV is a finite (R,Q)-biset, then the composition of (thisomorphism classes of)V andU is the (isomorphism class) ofV ×Q U . The identitymorphism IdP of thep-groupP is the class of the setP , with left and right action bymultiplication.

Let Fp,k denote the category ofk-linear functors fromCp,k to the categoryk-Mod ofk-modules. An object ofFp,k is called a biset functor (defined overp-groups, with valuesin k-Mod).

If F is an object ofFp,k, if P andQ are finitep-groups, and ifϕ ∈ HomCp,k(P,Q),

then the image ofw ∈ F(P) by the mapF(ϕ) will generally be denoted byϕ(w). Thecompositionψ ϕ of morphismsϕ ∈ HomCp,k

(P,Q) andψ ∈ HomCp,k(Q,R) will also

be denoted byψ ×Q ϕ.

2.2. Examples

Recall that this formalism of bisets gives a single framework for the usual operations oinduction, restriction, inflation, deflation, and transport by isomorphism via the follocorrespondences:

• If Q is a subgroup ofP , then let IndPQ ∈ HomCp,k(Q,P ) denote the setP , with left

action ofP and right action ofQ by multiplication.• If Q is a subgroup ofP , then let ResPQ ∈ HomCp,k

(P,Q) denote the setP , with leftaction ofQ and right action ofP by multiplication.

• If N P , andQ = P/N , then let InfPQ ∈ HomCp,k(Q,P ) denote the setQ, with left

action ofP by projection and multiplication, and right action ofQ by multiplication.• If N P , andQ = P/N , then let DefPQ ∈ HomCp,k

(P,Q) denote the setQ, with leftaction ofQ by multiplication, and right action ofP by projection and multiplication.

• If ϕ :P → Q is a group isomorphism, then let IsoQP = IsoQ

P (ϕ) ∈ HomCp,k(P,Q) de-

note the setQ, with left action ofQ by multiplication, and right action ofP by takingimage byϕ, and then multiplying inQ.

2.3. Sections and isomorphisms

Let (T ,S) and(T ′, S′) be sections of the groupP , and letx ∈ P be an element such th(T ,S) (xT ′, xS′). Let U denote the setS\T xT ′/S′, viewed as a(T /S,T ′/S′)-biset. If


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et

t ′ ∈ T ′, then there existst ∈ T ∩ xT ′ ands′ ∈ S′ such thatt ′ = txs′. The correspondencu : t ′S′ → tS is well defined, and it is a group isomorphism fromT ′/S′ to T/S. Moreover,the correspondence

∀(t, t ′) ∈ T × T ′, Stxt ′S′ ∈ S\T xT ′/S′ → tS.u(t ′S′) ∈ T/S

is an isomorphism of(T /S,T ′/S′)-bisets betweenU and IsoT/S

T ′/S ′(u).

2.4. Opposite bisets

If P andQ are finitep-groups, and ifU is a finite(Q,P )-biset, then letUop denote theopposite biset: as a set, it is equal toU , and it is a(P,Q)-biset for the following action

∀h ∈ Q, ∀u ∈ U, ∀g ∈ P, g.u.h (in Uop) = h−1ug−1 (in U).

This definition can be extended by linearity, to give an isomorphism

ϕ → ϕop: HomCp,k(P,Q) → HomCp,k

(Q,P ).

It is easy to check that(ϕ ψ)op = ψop ϕop, with obvious notation, and the functor

P → P, ϕ → ϕop

is an equivalence of categories fromCp,k to the dual category.

2.5. The functorP → RQ(P )

If P andQ are finitep-groups, ifV is a finitely generatedQP -module, ifU is a finite(Q,P )-biset, thenQU is a (QQ,QP)-bimodule, andQU ⊗QP V is a finitely generatedQQ-module. This definition can be extended bylinearity, and gives a structure of bisfunctor (with values inZ-Mod) to the correspondenceP → RQ(P ).

This functorial structure is well-behaved with respect to the bilinear forms〈 , 〉P , inthe following sense: ifP and Q are finitep-groups, ifa ∈ RQ(P ) and b ∈ RQ(Q), ifϕ ∈ HomCp,k

(P,Q), then

⟨b,ϕ(a)

⟩Q

= ⟨ϕop(b), a

⟩P.

2.6. Some idempotents inEndCp,k(P )

Let P be a finitep-group, and letN P . Then it is clear from the definitions that

DefPP/N InfPP/N = (P/N) ×P (P/N) = IdP/N .

It follows that the composition

ePN = InfPP/N DefPP/N


nce

ry

is an idempotent in EndCp,k(P ). It is the class of the setP/N , viewed as a(P,P )-biset by

left and right action given by projection onP/N , and multiplication inP/N . Moreover, ifM andN are normal subgroups ofP , then

eN eM = (P/N) ×P (P/M),

which is easily seen to be isomorphic toP/NM as(P,P )-biset. In other words

ePN eP

M = ePNM.

MoreovereP1 = IdP . The following lemma is well-known (and can be proved for insta

using [3, Lemme 8]).

2.7. Lemma. If N P , definef PN ∈ EndCp,k

(P ) by

f PN =

∑MPN⊆M

µP (N,M)ePM,

whereµP denotes the Möbius function of the poset of normal subgroups ofP . Thenthe elementsf P

N , for N P , are orthogonal idempotents ofEndCp,k(P ), and their sum is

equal toIdP .

Now if N P , the poset]1,N[P of normal subgroups ofP which are non-trivial andproperly contained inN is contractible, unless the action ofP onN is semi-simple. SinceP is a p-group, this implies that ifµP (1,N) = 0, thenN is central inP . In this casemoreoverµP (1,N) is equal to the Möbius functionµ(1,N) of the poset of all propenon-trivial subgroups ofN . This is zero ifN is not elementary abelian. This gives finall

f P1 =

∑N⊆Ω1Z(P )

µ(1,N)P/N,

whereΩ1Z(P) is the subgroup of the center ofP consisting of elements of order at mostp.

2.8. Faithful elements for biset functors

If F is an object ofFp,k, and if P is a finite p-group, then the idempotentf P1 of

EndCp,k(P ) acts onF(P). Its image

∂F (P ) = F(f P

1

)F(P)

is a direct summand ofF(P) ask-module: it will be called the set offaithful elements ofF(P), because any elementu ∈ F(P) which is inflated from a proper quotient ofP is suchthatF(f P

1 )u = 0. In the casek = Z andF = RQ, the module∂F (P ) has a basis overZconsisting of the faithful irreducible rational representations ofP .


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3. Bisets and genetic sections

3.1. Notation. Let P be a finitep-group, and(T ,S) be a genetic section ofP , with S = T .Then(T /S) has a unique central subgroup of orderp, and this subgroup is equal toS/S,for some well defined subgroupS of T . ThenP/S and P/S are (P,T /S)-bisets, andI denote byaT ,S the element of HomCp,k

(T /S,P ) defined by

aT ,S = P/S − P/S.

I denote bybT ,S the element of HomCp,k(P,T /S) corresponding to the opposite bisets,

bT ,S = S\P − S\P.

If (T ,S) = (P,P ) is the trivial genetic section ofP , I denote byaT ,S the image inHomCp,k

(1,P ) of a (P,1)-biset of cardinality 1, and bybT ,S the image in HomCp,k(P,1)

of a (1,P )-biset of cardinality 1.

The main theorem can now be stated as follows:

3.2. Theorem. LetF be a biset functor defined overp-groups, with values ink-Mod. Thenfor any finitep-groupP , and any setS of representatives of genetic sections ofP modulothe relation P , the map

IS =⊕

(T ,S)∈SF(aT ,S) :

⊕(T ,S)∈S

∂F (T /S) → F(P),

is a split injection ofk-modules, with left inverse

DS =⊕

(T ,S)∈SF(bT ,S) :F(P) →

⊕(T ,S)∈S

∂F (T /S).

4. Some combinatorial properties of origins and genetic sections

4.1. Lemma. LetP be a finitep-group, andQ be an origin inP , with associated genetisection(TQ,SQ), and corresponding simpleQP -moduleVQ. Then ifR is any subgroupof P , the multiplicitym(VQ,QP/R) of VQ in the permutation moduleQP/R is equal to

m(VQ,QP/R) = |Q : SQ|a + b,

where

a = cardx ∈ TQ\P/R | TQ ∩ xR ⊆ SQ

,

b = cardx ∈ TQ\P/R

∣∣ ∣∣(TQ ∩ xR)SQ/SQ

∣∣ = p, (TQ ∩ xR)SQ/SQ ⊆ Z(TQ/SQ).


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e

Proof. First, the result holds ifQ = P , for in this caseVQ = Q andTQ = SQ = P , thusm(VQ,QP/R) = 1, and moreovera = 1 andb = 0.

Suppose now thatQ is a proper subgroup ofP . Set(T ,S) = (TQ,SQ) andD = T/S.Then

〈VQ,QP/R〉P = ⟨IndP

T InfTD ΦD,QP/R⟩P

= ⟨ΦD,DefTD ResPT QP/R

⟩D

.

Now DefTD ResPT QP/R is isomorphic toQS\P/R asQD-module. In other words

DefTD ResPT QP/R ∼=⊕

x∈T \P/R

QS\T xR/R,

and theQD-moduleQS\T xR/R is isomorphic toQD/Ex , whereEx = (T ∩ xR)S/S,for eachx ∈ P : indeedS\T xR/R is a transitiveD-set, and the stabilizer ofSxR in D isequal toEx . It follows that

〈VQ,QP/R〉P =∑

x∈T \P/R

〈ΦD,QD/Ex〉D.

Now there are two cases: ifEx contains the unique central subgroupS/S of orderp of D,then the permutation moduleQD/Ex is inflated from a proper quotient ofD. Hence it is adirect sum of non-faithful irreducibleQD-modules. Hence

〈ΦD,QD/Ex〉D = 0

in this case. And ifEx does not containS/S, then it is either trivial, or non-central oorderp (this can only happen of course ifP is dihedral or semi-dihedral, hence ifp = 2).

If Ex is trivial, i.e. if T ∩ xR ⊆ S, then

〈ΦD,QD/Ex〉D = dimQ ΦD.

And if Ex is non-central of orderp in D (hence ifp = 2 andD is dihedral or semi-dihedral), then

〈ΦD,QD/Ex〉D = 〈QD/Ex,QD/Ex〉D = 〈ΦD,ΦD〉Dm(ΦD,QD/Ex).

But in this caseE = Ex is a basic subgroup ofD, andΦD is the corresponding simplQD-moduleVE (see [1, 3.7]). By [1, assertion (2) of Proposition 2.5],

m(VE,QD/E) = |ID(E,E)||ID(E,E)| = 1,

thus

〈ΦD,QD/Ex〉D = 〈ΦD,ΦD〉D.


e

This gives finally

〈VQ,QP/R〉P = a dimQ ΦD + b〈ΦD,ΦD〉Dwhere

a = cardx ∈ T \P/R | Ex = 1,b = card

x ∈ T \P/R | |Ex | = 2, Ex ⊆ Z(D)

On the other hand,

〈VQ,QP/R〉P = 〈VQ,VQ〉P m(VQ,QP/R) = 〈ΦD,ΦD〉Dm(VQ,QP/R),

hence

m(VQ,QP/R) = dimQ ΦD

〈ΦD,ΦD〉D a + b.

Now there are two cases: ifD is cyclic or generalized quaternion, then

dimQ ΦD = (p − 1)|D|/p = 〈ΦD,ΦD〉D(see [1, 3.7 and 5.5]), andQ = SQ in this case. Lemma 4.1 holds in this case. And ifD isdihedral or semi-dihedral, then dimQ ΦD = |D|/4 and〈ΦD,ΦD〉D = |D|/8, thus

dimQ ΦD

〈ΦD,ΦD〉D = 2.

But |Q : SQ| = 2 in this case, and this completes the proof.4.2. Lemma. LetQ be an origin inP , with associated section(TQ,SQ) and correspondingsimpleQP -moduleVQ. Then the multiplicitym(VQ,QP/SQ) is equal to|Q : SQ|, and thekernel of the projection mapQP/SQ → QP/SQ is isomorphic to a direct sum of|Q : SQ|copies ofVQ.

Proof. Let D = TQ/SQ. If D is cyclic or generalized quaternion, thenQ = SQ, Q = SQ,and

m(VQ,QP/Q) = |IP (Q,Q)||IP (Q,Q)| = 1,

and the result holds in this case, sinceVQ is the kernel of the projection mapQP/Q →QP/Q.

Suppose thatD is dihedral or semi-dihedral (hencep = 2), and letZ denote the uniqucentral subgroup of order 2 ofD. There is a unique faithful irreducibleQD-moduleΦD ,


the

ts

d

such that DefDD/Z ΦD = 0 (see [1, 3.12]). Any other irreducibleQD-moduleW is inflated

from D/Z, i.e. such thatW = InfDD/Z DefDD/Z W . Now QD/1 splits as

QD/1 = mΦD ⊕ D′

for somem ∈ N, whereD′ is a sum of simple modules which are inflated fromD/Z. Theabove considerations show thatD′ = InfDD/Z DefDD/Z QD = QD/Z. Taking dimensions inthe previous equality gives

|D| = mdimQ ΦD + |D|/2 = m|D|/4+ |D|/2,

and it follows thatm = 2, thus

QD/1 = 2ΦD ⊕ QD/Z.

Taking inflation fromD to TQ, and then induction toP gives

QP/SQ = 2VQ ⊕ QP/SQ,

thus

m(VQ,QP/SQ) = 2+ m(VQ,QP/SQ

).

SinceQ/SQ is a non-central subgroup of order 2 ofTQ/SQ, it follows thatQ ⊆ SQ, and|Q.SQ| > |Q| (actually|Q.SQ| = 2|Q| sinceQ ∩ SQ = SQ). SinceQ is basic, it followsthatIP ((Q.SQ),Q) = ∅, hence thatm(VQ,QP/(Q.SQ)) = 0.

Now if m(VQ,QP/SQ) = 0, it follows thatVQ appears as a direct summand ofkernelKQ of the projection map

QP/SQ → QP/(Q.SQ

).

The dimension ofKQ is equal to

dimQ KQ = |P : SQ|2

− |P : SQ|4

= |P : SQ|4

= dimQ VQ.

HenceKQ∼= VQ is an irreducibleQP -module. By [4, Proposition 4], it follows thatSQ is

a basic subgroup ofP , and in particular the groupW = NP (SQ)/SQ is cyclic or general-ized quaternion. ButW contains the groupTQ/SQ

∼= D/Z, which is dihedral. This cannohappen, and it follows thatm(VQ,QP/SQ) = 0, thusm(VQ,QP/SQ) = 2 in this case, awas to be shown.

Now the kernelLQ of the projection mapQP/SQ → QP/SQ has a direct summanisomorphic to a direct sum of 2 copies ofVQ. But

dimQ LQ = |P : SQ| − 1

2|P : SQ| = 2 dimQ VQ,

henceLQ∼= 2VQ, completing the proof.


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d

4.3. Corollary. Let (T ,S) be a non-trivial genetic section ofP , and letx ∈ P .

(1) If S ∩ xS ⊆ S, thenx ∈ T . In particular,T = NP (S).(2) The group(T ∩ xS)S/S cannot be a non-central subgroup of orderp of T/S.

Proof. There exists an originQ in P such that(T ,S) = (TQ,SQ). With the notation ofthe two previous lemmas, one has that

|Q : S|a + b = |Q : S|,

where

a = cardx ∈ T \P/S | T ∩ xS ⊆ S,b = card

x ∈ T \P/S

∣∣ ∣∣(T ∩ xS)S/S∣∣ = p, (T ∩ xS)S/S ⊆ Z(T/S)

,

Thusa 1, andb 0. Since|Q : S|(a − 1) + b = 0, it follows thatb = 0 anda = 1.Assertion (2) follows fromb = 0. For assertion (1), observe that the equalitya = 1 meansthat

∀x ∈ P, T ∩ xS ⊆ S ⇒ x ∈ T .

SetL = (T ∩ xS)S. ThenL/S is a subgroup ofT/S. If S ∩ xS ⊆ S, thenL/S ∩ S/S = 1.SinceL/S cannot be non-central of order 2 inT/S, it follows thatL/S = 1. ThusT ∩ xS ⊆S, andx ∈ T .

Corollary 4.3 leads to the following combinatorial characterization of genetic section

4.4. Proposition. Let P be a finitep-group, and let(T ,S) be a section ofP . Let ZP (S)

denote the subgroup ofP defined by

ZP (S)/S = Z(NP (S)/S

).

Then the following conditions are equivalent:

(1) the section(T ,S) is a genetic section ofP ;(2) the groupNP (S)/S has normalp-rank1, the groupT is equal toNP (S), and ifx ∈ P

is such thatSx ∩ ZP (S) ⊆ S, thenx ∈ NP (S).

Proof. Suppose first that(T ,S) is a genetic section ofP . Then eitherT = S = P , andassertion (2) holds trivially, or(T ,S) is a non-trivial genetic section. In this caseT =NP (S) by Corollary 4.3, thusNP (S)/S = T/S has normalp-rank 1. Letx ∈ P . SinceS/S = Ω1(ZP (S)/S), the hypothesesSx ∩ S ⊆ S andSx ∩ ZP (S) ⊆ S are equivalent, anassertion (2) holds by Corollary 4.3.


of

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f

t

ity

m-

Conversely, suppose that assertion (2) holds. Then eitherS = T = P , and (T ,S) isa genetic section ofP , or S is a proper subgroup ofP , thus also a proper subgroupT = NP (S). Let S be the subgroup ofP defined byS/S = Ω1(Z(T /S)).

Let Q/S be a basic subgroup ofT/S, such thatQ/S ∩ Z(T/S) = 1. Such a subgrouexists becauseT/S has a faithful irreducible rational representation. Then in particS ⊆ NP (Q) andQ ∩ S = S. Now if x ∈ IP (Q,Q), then

Sx ∩ S ⊆ Qx ∩ NP (Q) ⊆ Q ∩ S = S,

henceSx = S. ThusIP (Q,Q) ⊆ NP (S) = T , and in particularNP (Q) = NT (Q). Let I

denote the set of elementsx ∈ T such thatxS ∈ IT/S(Q/S,Q/S). If x ∈ I , then

Qx ∩ NP (Q) = Qx ∩ NT (Q) ⊆ Q,

thusx ∈ IP (Q,Q). Moreover,I generatesT , sinceIT/S(Q/S,Q/S) generatesT/S by[1, Lemma 7.2]. It follows that the groupTQ generated byIP (Q,Q) containsT , and iscontained inNP (S). HenceTQ = T = NP (S).

Now SQ is the largest normal subgroup ofT contained inQ, hence it is equal toS,sinceQ/S intersects the center ofT/S trivially. Hence(TQ,SQ) = (T ,S), andTQ/SQ hasnormalp-rank 1. ThusQ is an origin inP , and(T ,S) = (TQ,SQ) is a genetic section oP by [1, Proposition 7.4]. 4.5. Lemma. Let (T ,S) and (T ′, S′) be non-trivial genetic sections ofP . Suppose thathere existsx andy in P such that

S ∩ xS′ ⊆ S and S′ ∩ yS ⊆ S′.

Then(T ,S) P (T ′, S′).

Proof. Choose originsR and R′ in P such that(T ,S) = (TR,SR) and (T ′, S′) =(TR′, SR′). Consider the subgroupL = (T ∩ xS′).S/S of T/S. Since

L ∩ (S/S

) = ((T ∩ xS′).S ∩ S

)/S = (

T ∩ xS′ ∩ S).S/S = (

S ∩ xS′).S/S = 1,

the group L is trivial or non-central of order 2. By Lemma 4.1, the multiplicm(VR,QP/S′) is non-zero. SinceR is basic, it follows that|S′| |R|. By symmetrym(VR′,QP/S) = 0 and|S| |R′|.

Suppose that|R′| < |R|. Then|R′| |R|/p |S| |R′|. Thus|R′| = |R|/p = |S| inthis case, and this can only happen ifp = 2 andT/S is dihedral or semi-dihedral.

Since|R| = |R′|, the modulesVR andVR′ are not isomorphic. Hence there is a decoposition

QP/S = Q ⊕ 2VR ⊕ VR′ ⊕ W


n

l.

l

for someQP -moduleW , sincem(VR,QP/S) = 2 by Lemma 4.2. It follows that

|P : S| > 2|P : S|

4+ |P : R′|

2= |P : S|

2+ |P : R′|

2,

hence |R′| > |S| = |R|/2. It follows that |R′| |R|, contradicting the assumptio|R′| < |R|.

By symmetry, it follows that|R| = |R′|. Suppose thatVR ∼= VR′ . If T/S andT ′/S′ areboth cyclic or generalized quaternion, thenR = S andR′ = S′. SinceQ ⊕ VR ⊕ VR′ is adirect summand ofQP/S = QP/R, it follows that

|P : R| >(

1− 1

p

)|P : R| +

(1− 1

p

)|P : R′| =

(2− 2

p

)|P : R| |P : R|

sincep 2. This is impossible.Hencep = 2, and at least one ofT/S or T ′/S′, sayT/S, is dihedral or semi-dihedra

This implies in particular that|R : S| = 2.If m(VR′,QP/R) = 0, thenQ ⊕ VR ⊕ VR′ is a direct summand ofQP/R, and

|P : R| >1

2|P : R| + 1

2|P : R′| = |P : R|,

which is impossible. Thusm(VR′,QP/R) = 0.Now there are decompositions

QP/S = Q ⊕ 2VR ⊕ VR′ ⊕ W, QP/R = Q ⊕ VR ⊕ W ′

for someQP -modulesW andW ′, with m(VR,W ′) = m(VR′,W ′) = 0. Hence the kerneK of the projection map

QP/S → QP/R

has a direct summand isomorphic toVR ⊕ VR′ . But

dimQ K = 2|P : R| − |P : R| = |P : R|,

whereas

dimQ(VR ⊕ VR′) = 1

2|P : R| + 1

2|P : R′| = |P : R|.

HenceK ∼= VR ⊕ VR′ and

QP/S ∼= VR ⊕ VR′ ⊕ QP/R ∼= 2VR ⊕ QP/S

by Lemma 4.2. This shows thatVR′ appears as a direct summand ofQP/S. On theother hand,VR′ cannot appear as a direct summand ofQP/(R.S), sinceR′ is basic,


t

and |R.S| > |R′|. HenceVR′ is a direct summand of the kernelL of the projection mapQP/S → QP/(R.S). Since

dimQ L = 1

2|P : S| − 1

4|P : S| = 1

4|P : S| = 1

2|P : R| = 1

2|P : R′| = dimQ VR′,

it follows thatL ∼= VR′ .SinceL is a simpleQP -module, it follows thatS is a basic subgroup ofP , hence

thatNP (S)/S is cyclic or generalized quaternion. But this group containsT/S, which isdihedral. This cannot happen, and this contradiction shows thatVR

∼= VR′ , or equivalently(T ,S) P (T ′, S′), completing the proof. 4.6. Theorem. Let P be a finitep-group. If σ = (T ,S) and σ ′ = (T ′, S′) are geneticsections ofP , set

πσσ ′ = bT ,S aT ′,S ′ ∈ HomCp,k

(T ′/S′, T /S).

If πσσ ′ = 0, thenσ P σ ′.

(Recall that in this case, the simpleQP -modules corresponding toσ andσ ′ are isomor-phic.)

Proof. Suppose first thatσ is the trivial genetic section. Ifσ ′ is also trivial, there is nothingto prove since thenσ ′ = σ , henceσ ′

P σ . And if σ ′ is non-trivial, then

bT ,S aT ′,S ′ = P\P × P

(P/S′ − P/S′) = P\P/S − P\P/S′ = 0.

A similar argument, or the fact that(πσσ ′)op = πσ ′

σ shows thatπσσ ′ = 0 if σ ′ is trivial andσ

is not.Hence I can suppose thatσ andσ ′ are both non-trivial. Then

πσσ ′ = (

S\P − S\P ) × P

(P/S′ − P/S′).

Claim. Let (B,A) be a section ofP such thatA\P (P/S − P/S) is a non-zero elemenof HomCp,k

(T /S,B/A). Then there exists an elementx ∈ P such thatS ∩ Ax ⊆ S.

Indeed the hypothesis means thatA\P/S = A\P/S in HomCp,k(T /S,B/A). Hence

there existsx ∈ P such thatAxS = AxS, or equivalentlyxS ⊆ AxS, i.e. S ⊆ Ax.S. Since|S : S| = p, this means thatS ∩ Ax ⊆ S, proving the claim.

Thus if πσσ ′ = 0 in HomCp,k

(T ′/S′, T /S), then at least one of the elements

u = S\P (P/S′ − P/S′) or v = S\P (

P/S′ − P/S′)is non-zero. Ifu = 0, then there existsy ∈ P such thatS′ ∩ yS ⊆ S′. And if v = 0, thenthere existsy ∈ P such thatS′ ∩ y S ⊆ S′, which impliesS′ ∩ yS ⊆ S′. Similarly, since


t

an

t

nt.

(πσσ ′)op = πσ ′

σ = 0, it follows that there existsx ∈ P such thatS ∩ xS′ ⊆ S. By Lemma 4.5,it follows that(T ,S) P (T ′, S′). 4.7. Lemma. Let (T ,S) and (T ′, S′) be genetic sections ofP , and suppose tha(T ,S) P (T ′, S′). Then

(1) there exists a unique double cosetSxT ′ in P such thatSx ∩ T ′ ⊆ S′. In this case(T ,S) (xT ′, xS′);

(2) if x ∈ P , the group(Sx ∩ T ′)S′/S′ is not a non-central subgroup of orderp of T ′/S′.

Proof. If (T ,S) is trivial, then so is(T ′, S′), and there is nothing to prove. Hence I csuppose that(T ,S) and (T ′, S′) are both non-trivial, and choose originsR andR′ in P

such that(TR,SR) = (T ,S) and(TR′, SR′) = (T ′, S′).There exists a unique double cosetT xT ′ such that(T ,S) x(T ′, S′). Since the simple

QP -modulesVR andVR′ associated to(T ,S) and(T ′, S′) are isomorphic, it follows tha|R| = |R′|. Moreover,R/S ∼= R′/S′ andT/S ∼= T ′/S′.

By Lemmas 4.1 and 4.2, one has that

m(VR,QP/S′) = m(VR′,QP/S′) = |R′ : S′| = |R : S| = |R : S|a + b,

where

a = cardx ∈ T ′\P/S | T ′ ∩ xS ⊆ S′,b = card

x ∈ T ′\P/S

∣∣ ∣∣(T ′ ∩ xS)S′/S′∣∣ = p, (T ′ ∩ xS)S′/S′ ⊆ Z(T ′/S′).

Sincea 1 if (T ,S) P (T ′, S′), it follows thata = 1 andb = 0. Assertion (2) followsfrom b = 0. Sincea = 1, there is a unique double cosetSxT ′ in P such thatSx ∩ T ′ ⊆ S′.This is the case in particular if(T ,S) x(T ′, S′). ThusSxT ′ = T xT ′ is also the uniquedouble cosetT xT in P such that(T ,S) x(T ′, S′). 4.8. Lemma. Let (T ,S) be a genetic section ofP . Then

aT ,S = aT ,SfT/S

1 , bT ,S = fT/S

1 bT ,S.

Proof. SincebT ,S = (aT ,S)op and fT/S

1 = (fT/S

1 )op, the two assertions are equivaleMoreover if (T ,S) is the trivial genetic section, the result is trivial. And if(T ,S) is non-trivial, thenΩ1Z(T/S) = S/S, thus

fT/S

1 = (T /S)/(S/S) − (T /S)/(S/S

) = T/S − T/S.

It follows that

(P/S) × T/SfT/S

1 = P/S − P/S = aT ,S,

henceaT ,S = aT ,SfT/S

1 , sincefT/S

1 is an idempotent.


-

s

ent

he

4.9. Theorem. Let σ = (T ,S) and σ ′ = (T ′, S′) be genetic sections ofP , and sup-pose that(T ,S) P (T ′, S′). Let T xT ′ denote the unique double coset inP such that(T ,S) x(T ′, S′). Denote byϕσ

σ ′ the (T /S,T ′/S′)-bisetS\T xT ′/S′, viewed as an element ofHomCp,k

(T ′/S′, T /S). Then

πσσ ′ = bT ,S aT ′,S ′ = f

T/S

1 ϕσσ ′ = ϕσ

σ ′fT ′/S ′1 in HomCp,k

(T ′/S′, T /S).

Proof. If σ is trivial, so isσ ′, and the result is trivial in this case. Suppose then thatσ andσ ′ are non-trivial, and denote byR andR′ origins ofP such that(T ,S) = (TR,SR) and(T ′, S′) = (TR′, SR′). Here again|R| = |R′|, the groupsR/S andR′/S′ are isomorphic, awell as the groupsT/S andT ′/S′. In particular|S| = |S′|.

There are two steps in the proof:

Step 1.

Claim. The compositionS\P (P/S′ − P/S′) is zero inHomCp,k(T ′/S′, T /S).

Indeed otherwise, by the Claim of the proof of Theorem 4.6, there exists an elemx

in P such thatSx ∩ S′ ⊆ S′. Then the subgroupL = (Sx ∩ T ′).S′/S′ of T ′/S′ is such thatL ∩ S′/S = 1. ThusL is either trivial, or non-central of orderp in T ′/S′. By Lemma 4.1,the multiplicitym(VR′,QP/S) is non-zero. SinceR′ is basic, this implies

|R′| ∣∣S∣∣ = ∣∣S′∣∣ |R|,

hence|R′| = |S′| = |R|. Hence |R′ : S′| = p, and this can occur only ifp = 2 andT/S and T ′/S′ are dihedral or semi-dihedral. Moreover|R.S| = 2|R| > |R′|, thusm(VR′,QP/(R.S)) = 0. HenceVR′ appears as a direct summand of the kernelK of theprojection mapQP/S → QP/(R.S). This kernel has dimension

dimQ K = 1

2|P : S| − 1

4|P : S| = 1

4|P : S| = dimQ VR′ ,

henceK ∼= VR′ is irreducible. ThusS is basic, henceNP (S)/S is cyclic or generalizedquaternion. But this group containsT/S, which is dihedral. This contradiction proves tclaim.

Step 2. It follows from Step 1 that

πσσ ′ = S\P × P

(P/S′ − P/S′)

=∑

x∈T \P/T ′

(S\T xT ′/S′ − S\T xT ′/S′) in HomCp,k

(T ′/S′, T /S).

Moreover, forx ∈ P

SxS′ = SxS′ ⇔ S′ = (Sx ∩ S′).S′ ⇔ Sx ∩ S′ ⊆ S′.


,
In that case the subgroupL = (Sx ∩ T ′).S′/S′ of T ′/S′ is trivial or non-central of order 2hence trivial by Lemma 4.7. There is a single double cosetT x0T
′ of such elementsx in P ,and

πσσ ′ =

( ∑x∈T \(P−T x0T

′)/T ′

(S\T xT ′/S′ − S\T xT ′/S′)) + (

S\T x0T′/S′ − S\T x0T

′/S′).If x ∈ P − T x0T

′, thenSyS′ = SyS′ for all y ∈ T xT ′. It follows that

S\T xT ′/S′ = S\T xT ′/S′

as(T /S,T ′/S′)-bisets, and finally

πσσ ′ = S\T x0T

′/S′ − S\T x0T′/S′.

Now recall thatf T ′/S ′1 = T ′/S′ − T ′/S′. It clearly follows that

πσσ ′ = (S\T x0T

′/S′) ×T ′/S ′ f T ′/S ′1 = ϕσ

σ ′fT ′/S ′1 .

Exchangingσ andσ ′ now gives

πσ ′σ = ϕσ ′

σ fT/S

1 ,

and taking opposite bisets gives

πσσ ′ = f

T/S

1 ϕσσ ′,

as was to be shown.

5. Proof of Theorem 3.2

Let F be a biset functor defined overp-groups, with values ink-Mod. Let P be a finitep-group, and letS be a set of representatives of genetic sections ofP modulo the relation

P . Denote byIS the map

IS =⊕

(T ,S)∈SF(aT ,S) :

⊕(T ,S)∈S

∂F (T /S) → F(P).

It follows from Lemma 4.8 that the image of the mapF(bT ,S) is contained in∂F (T /S),and one can define a mapDS by

DS =⊕

F(bT ,S) :F(P) →⊕

∂F (T /S).

(T ,S)∈S (T ,S)∈S


r, seeto

ee

lf

up of

is-e this

Chooseσ = (T ,S) ∈ S. Then for anyu ∈ ∂F (T /S), and for any(T ′, S′) in S, differentfrom (T ,S), one has that

F(bT ′,S ′)F (aT ,S)(u) = 0

by Theorem 4.6. Moreover, by Theorem 4.9

F(bT ,S)F (aT ,S)(u) = F(ϕσ

σ

)F

(f

T/S

1

)(u) = u,

sinceF(fT/S

1 )(u) = u for u ∈ ∂F (T /S), and since moreover

ϕσσ = S\T/S = T/S = IdT/S .

It follows thatDS IS is the identity map, and Theorem 3.2 follows.

6. An application to the Dade group

The definitions and notation concerning the Dade group refer to [6] (in particula[6, Corollary 3.10] for the definition of the mapD(U) between Dade groups associateda bisetU ).

6.1. Theorem. Let P be a finitep-group. Denote byDt(P ) the torsion subgroup of thDade group ofP , and byT t (P ) the subgroup ofDt(P ) formed by the images of thtorsion endo-trivial modules.

(1) If Q is a non-trivialp-group of normalp-rank one, thenT t (Q) is equal to the kerneof the mapDefQQ/Z :Dt(Q) → Dt(Q/Z), whereZ is the unique central subgroup o

orderp in Q. It follows thatT t (Q) = D(fQ1 )Dt (Q).

(2) Let S be a set of representatives of genetic sections ofP , modulo the relation P .Then the map

IS =⊕

(R,S)∈STeninfPR/S :

⊕(R,S)∈S

T t (R/S) → Dt(P )

is a split injection.

Proof. Assertion (1) follows from Dade’s Theorem on the structure of the Dade grocyclic groups (see [9,10]), and of [8, Lemma 10.2] for the other cases.

Assertion (2) is not a direct consequence of Theorem 3.2, for the Dade groupnota biset functor overp-groups, because of the extra phenomenon of “Galois torsion,” described in [6, Section 3]. However, the proof of Theorem 3.2 can be adapted to takGalois torsion into account, in the following way:


waysctions

First observe that since the Dade group of the trivial group is trivial, one can alreplaceS by S − 1 in the statement of the theorem, i.e. suppose that the genetic seconsidered here are non-trivial ones.

It follows from assertion (1) that ifQ is a non-trivialp-group of normalp-rank one,and ifu ∈ T t (Q), then DefQQ/Z u = 0.

This means that the mapIS can also be defined by

IS =⊕

(R,S)∈SD(aR,S),

sinceD(P/S) = TeninfPR/S

DefR/S

R/S.

Moreover, ifv ∈ Dt(P ), then the elementu = D(bR,S)(v) is an element ofDt(R/S) in

the kernel of the deflation map DefR/S

R/S, henceu ∈ T t (R/S). This allows to define a map

DS :Dt(P ) →⊕

(R,S)∈ST t (R/S),

and this map will be a left inverse toIS .To see this, let(R′, S′) be an element ofS, and let(R,S) be any section ofP . If u is an

element ofDt(R′/S′), one has that

D(S\P)D(P/S′)(u) = DefresPR/S TeninfPR′/S ′ u = DefRR/S ResPR TenPR′ InfR

′R′/S ′ u

=∑

x∈R\P/R′γ|S:S∩xR′|D(S\RxR′/S′)(u).

Similarly

D(S\P)D(P/S′)(u) =

∑x∈R\P/R′

γ|S:S∩xR′|D(S\RxR′/S′)(u)

by [6, Proposition 3.10]. Hence

D(S\P)D(aR′ ,S ′)(u) =∑

x∈R\P/R′γ|S:S∩xR′|

(D(S\RxR′/S′)(u) − D

(S\RxR′/S′)(u)

).

If this is non-zero, then there existsy ∈ P such thatSyS′ = SyS′, or equivalently,S′ ∩Sy ⊆ S′.

Suppose now that(R,S) ∈ S. The same argument shows that ifD(S\P)D(aR,S)(u) isnon-zero, then there existsy ∈ P such thatS′ ∩ Sy ⊆ S′, and this impliesS′ ∩ Sy ⊆ S′.

Since

D(bR,S) = D(S\P) − D(S\P )

,


e-l

nd

it follows that if D(bR,S)D(aR′,S ′)(u) is non-zero, then there existsy ∈ P such thatS′ ∩Sy ⊆ S′.

Recall thatD(P/S′)(u) = 0 for u ∈ Dt(R′/S′). Hence

D(bR,S)D(aR,S)(u) = D(bR,S)D(P/S)(u).

By the above computation, this is equal to

∑x∈R\P/R′

(γ|S:S∩xR′|D(S\RxR′/S′)(u) − γ|S:S∩xR′|D

(S\RxR′/S′)(u)

).

If this is non-zero, then at least one of the following holds:

• There existsx ∈ P such that|S : S ∩ xR′| = |S : S ∩ xR′|, or equivalently,∣∣S : S∣∣ = p = ∣∣S ∩ xR′ : S ∩ xR′∣∣;henceS ∩ xR′ ⊆ S. This impliesS ∩ xS′ ⊆ S.

• There existsx ∈ P such thatSxS′ = SxS′, or equivalently,S ∩ xS′ ⊆ S.

In both cases, there existsx ∈ P with S ∩ xS′ ⊆ S.Now Lemma 4.5 shows that ifD(bR,S)D(aR′,S ′)(u) = 0, then (R,S) P (R′, S′),

hence(R,S) = (R′, S′) since both pairs are inS.In this case by Corollary 4.3, there is a unique double cosetRxR in P such thatS ∩

xS ⊆ S, namely the cosetR. It follows that

D(bR,S)D(aR,S)(u) = γ|S:S∩R|D(S\R/S)(u) − γ|S:S∩R|D(S\R/S

)(u)

= D(R/S)(u) − γ|S:S∩R|D(R/S

)(u) = u

sinceR/S = IdR/S andD(R/S)(u) = 0 for u ∈ Dt (R/S). It follows thatDS IS is theidentity map, and this completes the proof of the theorem.6.2. Conjecture. The mapIS of Theorem6.1 is an isomorphism, for any finitep-groupP ,and any setS of representatives of genetic sections ofP modulo the relation P .

By [8] and Theorem 6.1, this conjecture is equivalent to the following.

6.3. Conjecture. The torsion partDt(P ) of the Dade group ofP is isomorphic to(Z/2Z)nP ⊕ (Z/4Z)mP , wheremP is equal to the number of rational irreducible reprsentations ofP of generalized quaternion type, andnP is equal to the number of rationairreducible representations ofP whose type is

• cyclic of order at least3, or semi-dihedral, or generalized quaternion, if the groufield contains cubic roots of unity;


at

son

-a

s

plee

s

-,

theince

• cyclic of order at least3, or semi-dihedral, or generalized quaternion of orderleast16, if the ground field does not contain cubic roots of unity.

6.4. Remark. Conjectures 6.2 and 6.3 are known to be true in the following cases:

• for p = 2, by a result of J. Carlson and J. Thévenaz [7, Theorem 13.3];• for p = 2 and metacyclicp-groups, by a result of N. Mazza, independent of Carl

and Thévenaz’s result [11];• for cyclic, generalized quaternion, dihedral or semi-dihedral 2-groups, by [8];• for some other 2-groups, such asD8 ∗C4, D8 ∗D8, orD8 ∗Q8, by results of J. Théve

naz, and more generally for all (almost) extraspecialp-groups, by a result of N. Mazzand the author [5].

7. Rational biset functors and the torsion part of DΩ

The results of previous sections lead to the following definition.

7.1. Definition. Let F be a biset functor defined overp-groups, with values ink-Mod. ThefunctorF is called rational if for any finitep-groupP , there exists a setS of representativeof genetic sections ofP modulo the relation P such that the map

IS =⊕

(T ,S)∈SF(aT ,S) :

⊕(T ,S)∈S

∂F (T /S) → F(P)

is ak-module isomorphism.

7.2. Example. The word “rational” in the previous definition comes from the examk = Z, andF = RQ. Then for anyp-groupQ with normalp-rank 1, there is a uniqufaithful irreducible rational representationΦQ, and∂F (Q) ∼= Z, with basisΦQ. Moreover,any rational irreducible representationV of P is isomorphic to IndPT InfTT/S ΦT/S for asuitable genetic section(T ,S) of P , which can be chosen in a given setS of representativeof such sections. ThusV = RQ(aT ,S)(ΦT/S), and the mapIS is surjective for anyp-group,and any set of representativesS. Hence it is an isomorphism, andRQ is rational.

The following lemma shows that ifF is rational, then the mapIS is an isomorphismfor anysetS of representatives of equivalence classes of genetic sections ofP .

7.3. Lemma. Let F be a biset functor andP be ap-group. LetS andS ′ be sets of representatives of genetic sections ofP for the relation P . If the mapIS is an isomorphismthen the mapIS ′ is an isomorphism.

Proof. Showing that the mapIS ′ is an isomorphism is equivalent to showing thatmapDS ′ is injective, since this map is split surjective by Theorem 3.2. Equivalently, sIS is an isomorphism by hypothesis, this amounts to showing that the mapf = DS ′


nt

.

e sub-

sses

IS is injective. Letu = (uT ,S)(T ,S)∈S an element of Ker(f ), with uT,S ∈ ∂F (T /S). If(T ′, S′) ∈ S ′, then the component off (u) in ∂F (T ′/S′) is equal to

bT ′,S ′( ∑

(T ,S)∈SaT ,S(uT ,S)

).

Now bT ′,S ′ aT ,S is equal to zero by Theorem 4.6, unless(T ′, S′) P (T ,S). There is aunique such(T ,S) ∈ S, and for this one

bT ′,S ′ aT ,S(uT ,S) = ϕT ′,S ′T ,S (uT ,S)

by Theorem 4.9, sincef T/S

1 (uT ,S) = uT,S . It follows that this is equal to the compone

of f (u) in ∂F (T ′/S′). Hence this is zero, anduT,S = 0 sinceϕT ′,S ′T ,S is an isomorphism

Since for any(T ,S) ∈ S, there is a unique(T ′, S′) ∈ S ′ such that(T ′, S′) P (T ,S), thusuT,S = 0 for any(T ,S) ∈ S, andu = 0. 7.4. Proposition. Let F be a biset functor overp-groups, with values ink-Mod, let F ′ bea subfunctor ofF , and letM be anyk-module.

(1) The functorF is rational if and only ifF ′ andF/F ′ are rational.(2) If F is rational, thenHom(F,M) is rational.

The first assertion means that the class of rational biset functors is a Serrclass of all biset functors. In thesecond assertion, recall that Hom(F,M) is defined byHom(F,M)(P ) = Homk

(F(P),M

)for anyp-groupP , and

Hom(F,M)(ϕ)(α) = α F(ϕop)

for ϕ ∈ HomCp,k(P,Q) andα ∈ Hom(F,M)(P ).

Proof. Let P be a finitep-group, andS be a set of representatives of equivalence claof genetic sections ofP . Let F ′′ = F/F ′. The diagram

0 0

⊕(T ,S)∈S ∂F ′(T /S)

i′F ′(P )

⊕(T ,S)∈S ∂F (T /S)

iF (P )

⊕(T ,S)∈S ∂F ′′(T /S)

i′′F ′′(P )

0 0


s.nce

y

sen-

et for

is commutative, where the horizontal maps are the mapsIS for the corresponding functorBy the snake’s lemma, and sincei ′, i ′, andi ′′ are (split) injective, there is an exact seque

0 → Coker(i ′) → Coker(i) → Coker(i ′′) → 0.

Now saying thatF is rational is equivalent to saying that Coker(i) = 0 for anyP andS.Thus Coker(i ′) = Coker(i ′′) = 0, andF ′ andF ′′ are rational. Conversely, ifF ′ andF ′′ arerational, then Coker(i ′) = Coker(i ′′) = 0, and Coker(i) = 0, thusF is rational.

For the second assertion, denote byF the functor Hom(F,M), and observe that for angenetic section(T ,S) of P , one has thatbT ,S = a

opT ,S , and that(f T/S

1 )op = fT/S

1 . Applyingthe functor Homk(−,M) to the isomorphism

IS =⊕

(T ,S)∈SF(aT ,S) :

⊕(T ,S)∈S

∂F (T /S) → F(P)

gives the isomorphism

DS =⊕

(T ,S)∈SF (bT ,S) : F (P ) →

⊕(T ,S)∈S

∂F (T /S),

for the functorF . ThusF is rational. 7.5. Corollary. The functorDΩ

tors is rational.

Proof. This follows from the fact that, by [2], the image of the subfunctorR∗Q

=Hom(RQ,Z) of the dual Burnside functorB∗ is equal to the torsion subfunctorDΩ

torsof DΩ . HenceDΩ

tors is a quotient of Hom(RQ,Z). SinceRQ is a rationalZ-valued bisetfunctor, so is its dual, and any quotient of it.7.6. Corollary. LetP be a finitep-group. Then

DΩtors(P ) ∼= (Z/4Z)aP ⊕ (Z/2Z)bP ,

whereaP is equal to the number of isomorphism classes of rational irreducible repretations ofP whose type is generalized quaternion, andbP is the number of isomorphismclasses of rational irreducible representations ofP whose type is cyclic of order at least3or semi-dihedral.

Proof. This follows from the structure of the Dade group of groups of normalp-rank 1(see [8, Theorems 5.4, 6.3, 7.1]).

Note that Corollary 7.5 is actually more precise, since it gives a generating sDΩ

tors(P ) ∼= (Z/4Z)aP ⊕ (Z/2Z)bP : if Q has normalp-rank 1, then∂DΩtors(Q) is a sub-

group of the endo-trivial subgroup ofD(Q). In particular,DΩ(aT,S)(u) = TeninfPT/S(u)

for any genetic section(T ,S) of P , and anyu ∈ ∂DΩtors(T /S).


are ay can

4)

in

Now if Q = T/S is cyclic or generalized quaternion, then∂DΩtors(Q) is generated by

uT,S = ΩQ/1, and ifQ is semi-dihedral, then∂DΩtors is generated byuT,S = ΩQ/1 +ΩQ/R,

whereR is a non-central subgroup ofQ of order 2. The elementsvT ,S = TeninfPT/S(uT ,S),for (T ,S) in a setS of representatives of equivalence classes of genetic sections ofP forwhich T/S is cyclic of order at least 3, or semi-dihedral, or generalized quaternion,set of generators ofDΩ

tors(P ), and these elements are as linearly independent as thebe: the linear combination

∑(T ,S)∈S λT,SvT ,S is zero if and only if for any(T ,S) ∈ S, the

integerλT,S is a multiple of the order ofvT ,S , which is equal to 4 ifT/S is generalizedquaternion, and 2 otherwise.

References

[1] S. Bouc, The functor of rational representations forp-groups, Adv. Math. 186 (2004) 267–306.[2] S. Bouc, A remark on the Dade group and the Burnside group, J. Algebra 279 (2004) 180–190.[3] S. Bouc, Construction de foncteurs entre catégories deG-ensembles, J. Algebra 183 (1996) 737–825.[4] S. Bouc, A remark on a theorem of Ritter and Segal, J. Group Theory 4 (2001) 11–18.[5] S. Bouc, N. Mazza, The Dade group of (almost) extraspecialp-groups, J. Pure Appl. Algebra 192 (200

21–51.[6] S. Bouc, J. Thévenaz, The group of endo-permutation modules, Invent. Math. 139 (2000) 275–349.[7] J. Carlson, J. Thévenaz, The classification of torsion endo-trivial modules, preprint, 2003; Ann. of Math.,

press.[8] J. Carlson, J. Thévenaz, Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000) 303–335.[9] E. Dade, Endo-permutation modules overp-groups I, Ann. of Math. 107 (1978) 459–494.

[10] E. Dade, Endo-permutation modules overp-groups II, Ann. of Math. 108 (1978) 317–346.[11] N. Mazza, Modules d’endo-permutation,PhD thesis, Université de Lausanne, 2003.

biset functors and genetic sections for p-groups

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