On the length of the conjugacy classes of finite groups
If \(G\) is a finite group, then several results indicate that the structure of \(G\) is controlled to a large extent by arithmetical conditions on the degrees of the irreducible complex characters. In the paper under review the authors study the influence of the lengths of conjugacy classes of \(G\) on the structure of \(G\). There are surprising analogues to the character degree results. It turns out that in most cases arithmetical conditions on the class lengths restrict the structure of \(G\) more heavily than the same conditions imposed on the character degrees. For instance: A prime \(p\) does not divide \(| C|\) for each conjugacy class \(C\) of \(G\) if and only if the Sylow \(p\)-subgroup is contained in the center of \(G\). (This needs not the classification of finite simple groups while the analogue in the character degree case does (Ito-Michler Theorem).) To get the flavour of the many results the authors have the following may serve: Theorem 1: Suppose that \(| C|\) is squarefree for each conjugacy class \(C\) of \(G\). Then \(G\) is supersolvable and both \(| G/F(G)|\) and \(| F(G)'|\) are squarefree numbers. In particular, \(G'\leq F(G)\) and \(G/F(G)\) is cyclic. Moreover, the derived length of \(G\) is bounded by 3, the nilpotency class of \(F(G)\) by 2.
scientific article; zbMATH DE number 4136210
scientific article