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
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 × 6m−2 + 3, the Hm(G,Fp)≅Hm(NG(K∞(P)),Fp) for all
integers m ≧ 2.
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