Let G be a finite group and N_{G} denote the set of nontrivial proper normal subgroups of G. An element K of N_{G} is said to be ndecomposable if K is a union of n distinct conjugacy classes of G.
In this paper, we investigate the structure of finite groups G in which G′ is a union of three distinct conjugacy classes of G. We prove, under certain conditions, G is a Frobenius group with kernel G′ and its complement is abelian. Furthermore, we investigate the structure of finite groups G in which N_{G} ≠ ∅ and every element of N_{G} is ndecomposable, for a given n. When G is solvable or n=2,3,4, we determine the structure of such groups.
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