# Math Help - abelian factor group

1. ## abelian factor group

Suppose G is a group and $Z(G)=\{a\in G | ax=xa, \forall x\in G \}$. Show that $G/Z(G)$ is cyclic, then G is abelian

2. Originally Posted by GTK X Hunter
Suppose G is a group and $Z(G)=\{a\in G | ax=xa, \forall x\in G \}$. Show that $G/Z(G)$ is cyclic, then G is abelian
$G\slash Z(G)\,\, cyclic\, \Longrightarrow G\slash Z(G)=\left\,\,\,\mbox{for some}\,\,x\in G\,\Longrightarrow$

$\forall g\in G\,\,\exists\, n_g \in\mathbb{Z}\,\,s.t.\,\,gZ(G)=x^{n_g}Z(G)\Longrig htarrow\,\exists z_g\in Z(G)\,\,s.t.\,\,g=x^{n_g}z_g$

Well,now take two elements in G, express them as described above and prove they commute.

Tonio