Blocks With The Hyperfocal Subgroup Klein Four Group

Let p be a prime,and O a complete discrete valuation ring of characteristic 0 with an algebraically closed residue filed k of characteristic p.We assume that the fraction field K of O is big enough for any finite group considered below.Assume that the group G has a 2-block b with abelian defect group and a Klein four hyperfocal subgroup.R(G,b)is the set of generalized characters in b.Let G be a finite group containing G as a normal subgroup.Let n be the order of G,denote by Qn the n-th cyclotomic subfield of K.H is the subgroup of Gal(Qn/Q)consisting of those automorphism ? for which there exists a non-negative integer m such that ?(?)=?pm for all p'-roots of unity C in Qn.Clearly G-conjugation permutes the block of G and H permutes the block of G by action of coefficient.We assume that b is G×H-stable,then G×H permutes all maximal b-Brauer pair.Fix a maximal b-Brauer pair(D,bD).Denote by N the normalizer of maximal b-Brauer pair(D,bD)in G×H.Denote by N the normalizer of(D,bD)in G.Set e to be bD.We will show that there exists a bijection between R(G,b)and R(N,e)preserving the N-action.It is indicated that the Galois-Alperin-McKay conjecture holds for the 2-block with abelian defect group and a Klein four hyperfocal subgroup.