On the normal subgroup with exactly two \(G\)-conjugacy class sizes.
There have been many results over the last few years which relate the structure of groups to the sizes of conjugacy classes. The authors contribute to this study by proving the following interesting theorem: Let \(G\) be a finite group and \(R\) be a non-central normal subgroup of \(G\). Let \(N\) be a normal subgroup of \(G\) containing \(R\) such that each element of \(N\) has either \(1\) or \(m\) conjugates in \(G\), where \(m\) is some fixed natural number. Then \(N\) is nilpotent. By taking \(N\) to be the group \(G\), the old result of \textit{N. Itô} follows [Nagoya Math. J. 6, 17-28 (1953; Zbl 0053.01202)]. There are two comments which seem to the reviewer to be worth making. It is a straightforward result that in any transitive permutation group there is an element which acts without fixed points. In this paper there is no need to use the much stronger result that there is an element of prime-power order with this property, a result which depends on the classification of finite simple groups. The second comment is whether the condition that \(N\) contains a non-central Sylow subgroup is needed. The authors make no reference to this in the paper.
scientific article; zbMATH DE number 5647926
scientific article