On finite groups admitting a coprime automorphism of prime order
Consider an automorphism \(\sigma\) of order \(r\) for a finite group \(G\). If \(r\) is relatively prime to the order \(|G|\) of \(G\), \(\sigma\) is a coprime automorphism. From the paper: `The purpose of this paper is to prove the following theorem on a coprime automorphism without invoking the classification theorems (i.e. for the finite simple groups). Theorem. Let \(G\) be a finite group admitting a coprime automorphism \(\sigma\) of prime order. Suppose \(G\) satisfies the following two conditions. (1) There exists the unique \(\sigma\)-invariant Sylow subgroup for each prime divisor of \(|G|\). (2) \(G=HX\), where \(H=\{g \mid g=g^\sigma\), \(g \in G\}\) and \(X=\{g^{-1} g^\sigma \mid g \in G\}\). Then \(G\) is solvable.' As an understatement, the proof is a careful and discriminating analysis of a minimal counter-example and reflects patience and insight on the part of the author. For the convenience of the reader and to keep the article relatively self-contained, the author provides a compendium of fundamental results required in the development of the contradictions.
scientific article; zbMATH DE number 783851