Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative.

2010-11-29T08:24:59Z

Hello. I finding for correct proof of this statement: If group $G/Z(G)$ is cyclic, then $G$ is commutative.

Proof: $G/Z(G)$ is isomorphic to $In(G)$ that is cyclic, and than for every inner isomorphism $\gamma_a$ is $\gamma_a \gamma_b = \gamma_{ab} = \gamma_{ba} = \gamma_b \gamma_a$ and therefore for every $a,b \in G$ is $ab = ba$.

Is that proof complete, or I miss something? Thanks a lot for help.

Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative. - MathOverflow [closed]

Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative.