The set of orders of elements in a finite group
Let \(\omega(G)\) be the set of orders of elements in a finite group \(G\). It was conjectured by Praeger and Shi that for a given set \(\pi\) of positive integers, there exists 0, 1 or infinitely many mutually nonisomorphic groups \(G\) with \(\omega(G) = \pi\). In this paper this conjecture is falsified by proving: Theorem. Let \(G\) be a group with \(\omega(G) = \omega (L_ 3(5))\). Then \(G \simeq L_ 3(5)\) or \(L_ 3(5) : 2\), where the second group is the extension of \(L_ 3(5)\) by the contragredient automorphism. In the proof the author reduces the problem to the case where \(G\) is almost simple and the classification of finite simple groups can be applied.
scientific article; zbMATH DE number 738976