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Math Help - abelian factor group

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by GTK X Hunter View Post
    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<xZ(G)\right>\,\,\,\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
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