Finite groups with many conjugate elements
\textit{F. M. Markel} [J. Algebra 26, 69-74 (1973; Zbl 0274.20030)] conjectured that if \(G\) is a finite nonabelian solvable group in which no two distinct conjugacy classes have the same length, then \(G\) must be isomorphic to the symmetric group \(S_3\). Based on the analysis made by \textit{M. B. Ward} [Arch. Math. 54, 111-116 (1990; Zbl 0663.20021)] of these groups of order \(2^a3^b\) and using a battery of results about rational and \(\pi\)-rational groups, the author proves Markel's conjecture. A finite nonabelian group \(G\) is called a gdc-group if any two noncentral elements \(x\), \(y\) of \(G\) which have conjugacy classes of the same length are conjugate in \(G\). And \(G\) is called dc-group if there are no two distinct conjugacy classes in \(G\) of the same length. Stated in these terms, Markel's conjecture asserts that \(S_3\) is the unique nonabelian solvable dc-group. For finite (not necessarily solvable) groups \(G\), the author proves that a nonabelian group \(G\) is a gdc-group iff \(G\) is a dc-group. This result can be regarded as a dual result for the corresponding assertion about degrees of nonlinear characters obtained by \textit{Ya. Berkovich, D. Chillag} and \textit{M. Herzog} [Proc. Am. Math. Soc. 115, 955-959 (1992; Zbl 0822.20004)].
scientific article; zbMATH DE number 703238