Incomplete vertices in the prime graph on conjugacy class sizes of finite groups

Abstract

Given a finite group G, consider the prime graph built on the set of conjugacy class sizes of G. Denoting by \u3c00\u3c00 the set of vertices of this graph that are not adjacent to at least one other vertex, we show that the Hall \u3c00\u3c00-subgroups of G (which do exist) are metabelia