On group theory 5
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On group theory 5
Let \(\displaystyle{\left(G,\cdot\right)}\) be a group and \(\displaystyle{N}\) a normal subgroup of \(\displaystyle{\left(G,\cdot\right)}\), that is \(\displaystyle{N\trianglelefteq G}\) . Prove that there exists a subgroup \(\displaystyle{K}\) of the group \(\displaystyle{Z(G/N)}\) such that:
\[\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K\leq \left(Z(G/N),\cdot\right)}\]
\[\displaystyle{\left(Z(G)/Z(G)\cap N,\cdot\right)\simeq K\leq \left(Z(G/N),\cdot\right)}\]
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