On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Abstract

Let $G$ be a finite group and $A$ be a normal subgroup of $G$. We denote by $ncc(A)$ the number of $G$-conjugacy classes of $A$ and $A$ is called $n$-decomposable, if $ncc(A)=n$. Set ${\mathcal{K}}_G=\{ncc(A)|A⊲G\}$. Let $X$ be a non-empty subset of positive integers. A group $G$ is called $X$-decomposable, if ${\mathcal{K}}_G=X$. Ashrafi and his co-authors \cite{ash1,ash2,ash3,ash4,ash5} have characterized the $X$-decomposable non-perfect finite groups for $X=\{1,n\}$ and $n\le 10$. In this paper, we continue this problem and investigate the structure of $X$-decomposable non-perfect finite groups, for $X=\{1,2,3\}$. We prove that such a group is isomorphic to $Z_6,D_8,Q_8,S_4$, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup$(m,n)$ denotes the $m$th group of order $n$ in the small group library of GAP \cite{gap}.

8 pages