On finite $X$-decomposable groups for $X=\{1‎, ‎2‎, ‎3‎, ‎4\}$

Document Type : Research Paper


Shanghai University


Let $\mathcal {N}_G$ denote the set of all proper‎
‎normal subgroups of a group $G$ and $A$ be an element of $\mathcal‎
‎{N}_G$‎. ‎We use the notation $ncc(A)$ to denote the number of‎
‎distinct $G$-conjugacy classes contained in $A$ and also $\mathcal‎
‎{K}_G$ for the set $\{ncc(A)\ |\ A\in \mathcal {N}_G\}$‎. ‎Let $X$ be‎
‎a non-empty set of positive integers‎. ‎A group $G$ is said to be‎
‎$X$-decomposable‎, ‎if $\mathcal {K}_G=X$‎. ‎In this paper we give a‎
‎classification of finite $X$-decomposable groups for $X=\{1‎, ‎2‎, ‎3‎,


Main Subjects