# How to prove that Z(G) is not a maximal subgroup of G, where G is an arbitrary group? [duplicate]

How to prove that Z(G) is not a maximal subgroup of G, where G is an arbitrary group?

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Hint: Consider some $x\in G$ with $x\not\in Z(G)$. What can you say about $C_G(x)$? – Tobias Kildetoft Mar 18 at 19:32

## marked as duplicate by Tara B, Henry T. Horton, Cameron Buie, rschwieb, ThomasMar 18 at 20:25

Hint 1: If a maximal subgroup $M$ is normal in $G$, then $G/M$ is cyclic.
Hint 2: If $G/Z(G)$ is cyclic, then $G$ is abelian.