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

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- Oct 21st 2009, 10:56 AMGTK X Hunterabelian factor group
Suppose G is a group and $\displaystyle Z(G)=\{a\in G | ax=xa, \forall x\in G \}$. Show that $\displaystyle G/Z(G)$ is cyclic, then G is abelian

- Oct 21st 2009, 11:52 AMtonio
$\displaystyle G\slash Z(G)\,\, cyclic\, \Longrightarrow G\slash Z(G)=\left<xZ(G)\right>\,\,\,\mbox{for some}\,\,x\in G\,\Longrightarrow$

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

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

Tonio