Extensions of several coprime results to good action case

Abstract

Let [Formula: see text] and [Formula: see text] be groups where [Formula: see text] acts on [Formula: see text] by automorphisms. We say “the action of[Formula: see text] on[Formula: see text] is good” if the equality [Formula: see text] holds for any subgroup [Formula: see text] of [Formula: see text] and for any [Formula: see text]-invariant subgroup [Formula: see text] of [Formula: see text]. It is straightforward that every coprime action is a good action. In this work, we extend some results due to Ward, Gross, Shumyatsky, Jabara and Meng and Guo under coprime action to good action.