Decomposability of Finite Groups

Let G be a finite group and A is a normal subgroup of G. We denote by ncc (A) the number of conjugacy classes of A in G and A is called n-decomposable, if ncc (A)=n. Set KG =. Let X be a non-empty subset of positive integers. A group G is called X-decomposable, if KG = X.In this paper we characterize the {1, 3, 4} – decomposable finite non-perfect groups. We prove that such a group is isomorphic to Small Group (36, 9) the 9th group of order 36 in the small group library of GAP, a metabelian group of order, in which n is odd positive integer and is a Mersenne prime or a metabelian group of order, where and is a Mersenne prim.2000 mathematics Subject Classification: 20E34,20D10.