Existence of a normal subgroup of finite index

#group_theory

Let $N$ be a finite normal subgroup of $G$ . Prove there is a normal subgroup $M$ of $G$ such that $[G : M]$ is finite and $n m = m n$ for all $n \in N$ and $m \in M$ .

Hint: You may use the fact that the centralizer $C (h) := {g \in G ∣ g h g^{- 1} = h}$ is a subgroup of $G$ .)

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Let $N$ be a finite normal subgroup of $G$. Prove there is a normal subgroup $M$ of $G$ such that $[G:M]$ is finite and $nm=mn$ for all $n\in N$ and $m\in M$.

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\noindent ({\itshape Hint:} You may use the fact that the centralizer $C(h):=\{g\in G\mid ghg^{-1}=h\}$ is a subgroup of $G$.)