On finite \(p\)-groups which have only two conjugacy lengths
Let \(G\) be a nonabelian finite group. If all noncentral conjugacy classes of \(G\) have the same size \(n\), then, according to \textit{N. Itô} [Nagoya Math. J. 6, 17-28 (1953; Zbl 0053.01202)], \(n=p^\alpha\) for some prime \(p\) and \(G=P\times A\), where \(P\) is a Sylow \(p\)-subgroup of \(G\) and \(A\) is Abelian. Hence, the further study of such groups is reduced to that of \(p\)-groups. Itô has proved that \(G\) has an Abelian normal subgroup \(A\) such that \(G/A\) is of exponent \(p\) (through many years Isaacs proved that even \(\exp(G/Z(G))=p\)). Itô wrote: ``It may be of use to investigate whether there exists a group of type \((p^\alpha,1)\) ... of arbitrary high class''. As Isaacs proved, if \(p=2\) then \(\Phi(G)\leq Z(G)\), i.e., \(G\) is of class \(2\). It follows from Isaacs' result that \(G'\) is elementary Abelian. The author proves in this note that if \(p>2\), then the class of \(G\) does not exceed \(3\). The proof is highly computational.
scientific article; zbMATH DE number 1760710