Extensions of several coprime results to good action case

TL;DR

This paper generalizes several known results about coprime group actions to a broader class called good actions, which include coprime actions as a special case, thereby expanding the applicability of these results.

Contribution

It extends key theorems from coprime actions to the more general setting of good actions in group theory.

Findings

Extended results of Ward, Gross, Shumyatsky, Jabara, Meng, and Guo to good actions.

Showed that properties previously known for coprime actions hold under the broader good action condition.

Demonstrated that good actions encompass coprime actions as a special case.

Abstract

Let $G$ and $A$ be groups where $A$ acts on $G$ by automorphisms. We say "\textit{the action of $A$ on $G$ is good}" if the equality $% H=[H,B]C_H(B)$ holds for any subgroup $B$ of $A$ and for any $B$-invariant subgroup $H$ of $G$. It is straightforward that every coprime action is a good action. In the present work we extend some results due to Ward, Gross, Shumyatsky, Jabara, and Meng and Guo under coprime action to good action.