Some special coprime actions and their consequence

TL;DR

This paper investigates how coprime group actions influence the structure of the group, providing bounds on p-length and insights into the embedding of A-invariant p-subgroups under specific conditions.

Contribution

It introduces new bounds for the p-length of groups under coprime actions and explores the embedding properties of A-invariant p-subgroups with additional assumptions.

Findings

Bounds for p-length in terms of the order of A

Conditions for embedding of A-invariant p-subgroups

Impact of fixed point subgroup order on group structure

Abstract

Let a group $A$ act on the group $G$ coprimely. Suppose that the order of the fixed point subgroup $C_G(A)$ is not divisible by an arbitrary but fixed prime $p$. In the present paper we determine bounds for the $p$-length of the group $G$ in terms of the order of $A$, and investigate how some $A$-invariant $p$-subgroups are embedded in $G$ under various additional assumptions.