Vol. 102, No. 1, 1982

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An affirmative answer to Glauberman’s conjecture

Masahiko Miyamoto

Vol. 102 (1982), No. 1, 89–105
Abstract

Let G be a finite group and P be a Sylow p-subgroup of G for a prime p. The following question is raised by G. Glauberman.

Question 16.8. Does there exist a function f from the positive integers i to the positive integers such that

Hi(G,Fp ) ∼= Hi(NG (K ∞ (P )),Fp) whenever  p ≧ f(i)?

Here K denotes the section conjugacy functor constructed by G. Glauberman and Fp denotes the finite field consisting of p elements and by forgetting its multiplicative structure, we consider it as a trivial G-module.

In relation to the above conjecture, he proved the case i = 1 and D. F. Holt has recently proved f(2) 11. The purpose of this paper is to provide an affirmative answer to the question.

Theorem C. If p 12 × 6m2 + 3, the Hm(G,Fp)Hm(NG(K(P)),Fp) for all integers m 2.

Mathematical Subject Classification 2000
Primary: 20J06
Milestones
Published: 1 September 1982
Authors
Masahiko Miyamoto
Departament of Mathematics
University of Tsukuba
Tsukuba 305-8571
Japan