Nonsolvable groups with an automorphism inverting many elements
Let G be a finite group, let \(\alpha\in Aut(G)\) and \(T(\alpha)=\{g\in G|\alpha (g)=g^{-1}\}\). \textit{H. Liebeck} and \textit{D. MacHale} [Math. Z. 124, 51-63 (1972; Zbl 0213.305)] have shown that if \(| T(\alpha)| >| G|\), then G is abelian. The present paper is concerned with the case \(| T(\alpha)| >| G| /4\). It is proved that the only simple nonabelian group satisfying this condition is \(A_ 5\). In this case \(\alpha\) can be choosen to be the inner automorphism induced by an involution and \(| T(\alpha)| =4| A_ 5| /15\). As a corollary, the following sufficient condition for solvability is obtained: G is solvable if it has an automorphism \(\alpha\) such that \(| T(\alpha)| >4| G| /15.\) The main result of the paper is the characterization of the nonsolvable groups G having an automorphism \(\alpha\) with \(| T(\alpha)| >| G| /4\). Such a group is isomorphic to the direct product \(A_ 5\times A\), where A is abelian, \(A\subset T(\alpha)\). Moreover, \(| T(\alpha)| =4| G| /15.\) It seems that the techniques employed in this paper could be used in corresponding problems concerning finite groups having an automorphism which squares of cubes relative large proportions of the elements of the group.
scientific article; zbMATH DE number 4051871