Some remarks on conjugate closed groups. (Q644670)

The authors call a group \(G\) a CC-group if for every normal subgroup \(H\) of \(G\) and every element \(x\) of \(H\), the conjugacy classes of \(x\) in \(G\) and \(H\) coincide. They present a number of simple properties of CC-groups. Images and normal subgroups of CC-groups are CC-groups. Abelian groups, simple groups and the direct product of two CC-groups are CC-groups. Also clearly an Abelian normal subgroup of a CC-group is central, so soluble CC-groups are Abelian and hence so are finite CC-groups with cyclic Sylow 2-subgroups. The authors claim as their main result that a finite semisimple perfect CC-group \(G\) is a direct product of simple groups, something that is almost immediate from the normality in \(G\) of its components and the `Schreier conjecture'.