Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative. - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-20T09:01:14Z http://mathoverflow.net/feeds/question/47650 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47650/proof-that-if-group-g-zg-is-cyclic-then-g-is-commutative Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative. tomas.lang 2010-11-29T08:24:59Z 2010-11-29T08:24:59Z <p>Hello. I finding for correct proof of this statement: If group $G/Z(G)$ is cyclic, then $G$ is commutative.</p> <p>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 $ab = ba$.</p> <p>Is that proof complete, or I miss something? Thanks a lot for help.</p>