Finite groups with an automorphism that is a complete mapping

TL;DR

This paper proves that finite groups with an automorphism making the map g↦gα(g) bijective must be solvable, revealing a structural property related to automorphisms and group solvability.

Contribution

It establishes a new solvability criterion for finite groups based on the existence of a special automorphism with a bijective associated map.

Findings

Finite groups with such automorphisms are necessarily solvable.

The automorphism induces a bijective map g↦gα(g).

This property characterizes a class of solvable groups.

Abstract

We show that a finite group $G$ admitting an automorphism $\alpha$ such that the function $G\rightarrow G$, $g\mapsto g\alpha(g)$, is bijective is necessarily solvable.