Finite groups with coprime fixed-point-free automorphisms and applications

TL;DR

This paper investigates finite groups with a specific automorphism property resembling elementary abelian p-groups, proving their solvability and characterizing certain symmetric Cayley graphs derived from them.

Contribution

It introduces a class of finite groups with a unique automorphism behavior and proves their solvability, also characterizing related edge-transitive Cayley graphs.

Findings

Such groups are solvable.

Characterization of normal edge-transitive Cayley graphs as complete multipartite graphs.

Provides structural insights into automorphism actions on finite groups.

Abstract

We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$. In this paper, we show that such groups are solvable. As an application, we characterise a class of normal edge-transitive Cayley graphs of $G$, which are complete multipartite graphs.