Camina groups with few conjugacy classes

Abstract

Let \(G\) be a finite group having a proper normal subgroup \(K\) such that the conjugacy classes outside \(K\) coincide with the cosets of \(K\). The subgroup \(K\) turns out to be the derived subgroup of \(G\), so the group \(G\) is either abelian or Camina. Hence, we propose to classify Camina groups according to the number of conjugacy classes contained in the derived subgroup. We give the complete characterization of Camina groups when the derived subgroup is made up of two or three conjugacy classes, showing that such groups are all Frobenius or extra-special.