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Let $G$ be a finite group and ${\cal N}_G$ denote the set of non-trivial proper normal subgroups of $G$. An element $K$ of ${\cal N}_G$ is said to be $n$-decomposable 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 ${\cal N}_G\neq \emptyset$ and every element of ${\cal N}_G$ is $n$-decomposable, for a given $n$. When $G$ is solvable or $n=2,3,4$, we determine the structure of such groups.
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