On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes
نویسندگان
چکیده
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 KG = {ncc(A)|A⊳ G}. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X . Ashrafi and his co-authors [1,2,3,4,5] have characterized the X-decomposable nonperfect 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 Z6,D8,Q8,S4, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m,n) denotes the mth group of order n in the small group library of GAP [11].
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