# abelian factor group

• October 21st 2009, 11:56 AM
GTK X Hunter
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
• October 21st 2009, 12:52 PM
tonio
Quote:

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