Invariant complements normalized by the fixed point subgroup

Abstract

Abstract. Let A and G be finite groups and assume that A acts on G by automorphisms with (|A|, |G|) = 1. Let CG(A) be the fixed point subgroup and fix a prime number p. We prove that if G possesses an A-invariant p-complement that is normalized by CG(A), then G is p-solvable. We also extend Frobenius criterion for p-nilpotency, in the coprime action setting, for the primes 2 and 3.