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Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $Z(N)$ be a center of $N$, then I want to prove $Z(N)$ is normal in $G$. [I know $Z(G)$ is normal in $G$, but is this true?]

Following are my trails; Recall the definition of $Z(N) = \{ z \in N | zn=nz, \phantom{1} \forall n \in N\}$, and to show $Z(N)$ is normal in $G$, $\forall g \in G, z \in Z(N)$, $gzg^{-1} \in Z(N)$. i.e., for $h\in N$, $hgzg^{-1} = gzg^{-1} h$

But I cannot prove the last equation. If $g \in N$ then everything seems fine but $g \in N -G$...

Shaun
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phy_math
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    kabenyuk has shown you how to prove this, but it would be much more useful in the long term if you learnt about characteristic subgroups. The centre of a group is not only normal, it is a characteristic subgroup. And a characteristic subgroup of a normal subgroup of $G$ is always normal in $G$. – Derek Holt Sep 19 '21 at 08:03

1 Answers1

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Since $h\in N$ and $N$ is a normal subgroup of $G$: $hgzg^{-1} = g\cdot g^{-1}hg\cdot z\cdot g^{-1}=g\cdot z\cdot g^{-1}hg\cdot g^{-1}=gzg^{-1} h$

kabenyuk
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