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It would be useful, where possible, to have relations for Z-cohomological Mackey functors, but integral Brauer relations are hard to study. Integral Brauer relations

have been investigated in [14], [21] and [28]. It is in general very hard to show when two Z[G]-permutation lattices are isomorphic, it is remarked in [21] that one may show that if for some finite groupGandH1, H2 6Gwe haveZp[G/H1] =Zp[G/H2]

for all p then we may conclude that there is an integer n such that Z[G/H1]⊕n =

Z[G/H2]⊕n but that this need not hold forn= 1. One way around this is to note that the evaluations of a Z-Mackey functor M which only takes values in finitely generated abelian groups are entirely determined by the completed functorsM_p = Mp⊗Zp aspranges over all primes. Instead of Brauer relations overZtherefore, it is sensible to consider everywhere local Brauer relations. More formally let K_p(−) denote the kernel of the map from the Burnside functor to the Zp Representation

ring functor then an everywhere local Brauer relation for G is an element θ ∈ ∩_pK_p(G). Such relations will result in non-trivial isomorphisms on the evaluations of Z-cohomological Mackey functors such as the fixed point functor for a Z[G]- moduleM.

We will show that using the results of the previous chapters one may rapidly get results describing this kernel.

To begin to describe everywhere local relations we will rephrase the problem in terms of Green functors with inflation and use the results of chapter three. To that end we will require the following definition

Definition 6.3.1. LetG be a finite group, and let a(Z[G]) be the ring of integral representations, we will define the genus ring Γ(G) ofGto be the quotient ofa(Z[G]) by the ideal generated by formal differences of isomorphism classes [A]−[B] whenever

Aand B are in the same genus, that isZp⊗ZA∼=Zp⊗ZB for all primes p.

Letg(M) denote the genus containing aZ[G]-moduleM in Γ(G), then equip- ping Γ(−) with the maps Ind_G/H : g(M) 7→ g(IndG/H(M)), ResG/H : g(N) 7→ g(Res_G/H(N)) and Inf_G/N :g(L) 7→g(Inf_G/N(L)) makes it into a GFI. Our aim is then to describe the kernel GFI KΓ of the map of GFIs:

mΓ(G) :b(G)→Γ(G) [H]7→g(IndG/H(1))

Lemma 6.3.2. The class of coprimordial groups C(Im(mΓ)) is precisely the class

of groups which arep-hypo-elementary for some primep.

Proof. The kernel of restriction functorK_Im(m

Γ)(−) is equal to Q0

pKIm(mp)(−) whose

evaluation is non-trivial whenever a group is p-hypo-elementary for some prime

Lemma 6.3.3. Let q be a prime. The class of primordial groupsP(Im(mΓ)q)is the

class of groups which are (p, q)-Dress for at least one prime p.

Proof. We have the containment P(Im(mΓ)q) ⊆ ∪pDp,q by Lemma 3.3.4. Now we

note that ifG is not primordial of Im(mΓ)q then it is not primordial for Im(mp)q

for any primep. The result follows.

Now we may apply the results of Chapter three to describe for which non- primordial finite groupsG the primitive quotient PrimKΓ(G) is non-trivial.

Theorem 6.3.4. Let G be a finite group that is not a (p, q)-Dress group for any prime numbers p and q. Then:

(a) if every proper quotientQofGispQ-hypo-elementary for some primepQ, then

PrimKΓ(G)∼=Z;

(b) if there exists a fixed prime number q such that the following statements hold:

• any proper quotientQ ofGis a (pQ, q)-Dress group for at least one prime

pQ, and

• any Q which is (pH, q)-Dress and (p2, l)-Dress for l 6= q is pH-hypo- elementary, and

• at least one proper quotient Qthem is not hypo-elementary.

Then PrimKΓ(G)∼=Z/q

n

Z for some natural number n>1;

(c) if there exists a proper quotient of G that is not a (p, q)-Dress group for any prime numbers p and q, or if there exist pairs of prime numbers p1, q1 and p2, q2 with theqi distinct and, for i= 1 and 2, a proper quotient of G that is a non-pi-hypo-elementary (pi, qi)-Dress group, thenPrimKΓ(G) is trivial.

Moreover, in all cases,PrimKΓ(G) is generated by any element of KΓ(G)⊆b(G) of

the form[G/G] +P

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