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

摘要

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 K_G = {ncc(A)|A △G} 2. K_G = X. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if κ_G = X. Ashrafi and his co-authors have characterized the X-decomposable non-perfect finite groups for X = [1,n] and n ≤ 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, 84, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of order n in the small group library of GAP.