Fitting Subgroup and Nilpotent Residual of Fixed Points

Abstract

Let q be a prime and A an elementary abelian group of order at least q 3 acting by automorphisms on a finite q ′ -group G. It is proved that if |γ ∞ (C G (a))| ≤ m for any a ∈ A # , then the order of γ ∞ (G) is m-bounded. If F (C G (a)) has index at most m in C G (a) for any a ∈ A # , then the index of F 2 (G) is m-bounded.1991 Mathematics Subject Classification. 20D45.

Nilpotent residual and fitting subgroup of fixed points in finite groups

“…In the present article we extend the results obtained in [4] to the case where A is not necessarily abelian.…”

“…Thus, A is a finite group of prime exponent q and order at least q 3 acting on a finite q ′ -group G in such a manner that F (C G (a)) has index at most m in C G (a) for any a ∈ A # . We wish to show that F 2 (G) has m-bounded index in G. If A contains an abelian subgroup of order p 3 , the result is immediate from [4]. Therefore without loss of generality we assume that all subgroups of order p 3 in A are non-abelian.…”

“…Recently, in [4] it was proved that if A is an elementary abelian group of order at least q 3 acting by automorphisms on a finite q ′ -group G and if |γ ∞ (C G (a))| ≤ m for any a ∈ A # , then the order of γ ∞ (G) is m-bounded. If F (C G (a)) has index at most m in C G (a) for any a ∈ A # , then the index of F 2 (G) is m-bounded.…”

“…Recently, in [4] it was proved that if A is an elementary abelian group of order at least q 3 acting by automorphisms on a finite q ′ -group…”