Nilpotent residual and fitting subgroup of fixed points in finite groups

Abstract

Abstract Let q be a prime and A a finite q-group of exponent q acting by automorphisms on a finite q ′ {q^{\prime}} -group G. Assume that A has order at least q 3 {q^{3}} . We show that if γ ∞ ⁢ ( C G ⁢ ( a ) ) {\gamma_{\infty}(C_{G}(a))} has order at most m for any a ∈ A # {a\in A^{\#}} , then the order of γ ∞ ⁢ ( G ) {\gamma_{\infty}(G)} is bounded solely in terms of m. If the Fitting subgroup of C G ⁢ ( a ) {C_{G}(a)} has index at most m for any a ∈ A # {a\in A^{\#}} , then the second Fitting subgroup of G has index bounded solely in terms of m.