The $p$-period of finite group

Abstract

If r is ny finite group, the p-period of is defined to be the least positive integer q such that the cohomology groups fl(r, A) nd/+(r, A) hve isomorphic p-primary components for ll i nd 11 A [1, Ch.XII, Ex. 11].This is equivalent to the statement that/(, Z) hs n element of order p, the highest power of p dividing the order of r [1, Ch.XII, Ex. 11].I will sy that q is p-period for if it is multiple of he p-period.The ordinary period of the cohomology of is, of course, the least common multiple of ll the p-periods.It is known [1, Ch.XII, Ex. 11] that the p-period will be finite if nd only if the p-sylow subgroup of r is either cyclic or a generalized quternion group.The purpose of this pper is to give simple group- theoretic interpretation of the p-period of .The methods used here also give cohomological generalization of Grfin's second theorem [3, Ch.V, Th. 6].This will be presented in the Appendix since it is not needed in proving Theorems 1 nd 2. THEOREM 1.If the 2-sylow subgroup ofis cyclic, the 2-period is 2. If he 2-sylow subgroup of r is a (generalized) quaernion group, the 2-period is 4. THEOREM 2. Suppose p is odd and the p-sylow subgroup of r is cyclic.Let r be a p-sylow subgroup, and le be he group of auomorphisms of r induced by inner automorphisms of r.Then the p-period of r is wice the order of .The group is, of course, isomorphic to N(-)/C(r) where N and C denote the normlizer nd centralizer, respectively.Before proving these theorems, I will review some fcts bout the cohomol- ogy of groups.Suppose h'p---.r is monomorphism of finite groups.Then h induces mp of cohomology h*'(r, A)---. [I(p, A).Here A is r-module and so cn be regarded as p-module by means of h.This mp h* is defined as follows.Let W be Tte complex (or complete resolu- tion in the terminology of [1, Ch.XII, 3]) for v. Then p acts on W through h, nd W is p-free since h is monomorphism.Thus W is lso Tte com- plex for p.The map h* is now defined to be the mp of cohomology induced by the inclusion Hom(W, A) Hom(W, A).In cse h is n inclusion mp, h* is just the mp i(p, r) of [1, Ch.XII, 8].Suppose r H, x e H nd h'xrx---r is given by h(y)= x-yx.Then h*'fl*(v, A)-- *(xrx -, A) is just the mp c of [1, Ch.XII, 8].Let r' be subgroup of r, nd x n element of r.Then there re two ob- vious monomorphisms i, f, r' xr'x ---r', nmely, i(y) y nd f, (y)

Locations

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