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Math Help - cyclic

  1. #1
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    cyclic

    Prove that if G/Z(G) is cyclic, then G is abelian. (If G/Z(G) is cyclic with generator xZ(G), show that every element of G can be written in the form x^az for some integer a and some z in Z(G))

    Z(G) = { g in G | gx = xg for every x in G}
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  2. #2
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    Let \left< x\text{Z}(G) \right> = G/\text{Z}(G). If \alpha \in G then form \alpha \text{Z}(G) \in G/\text{Z}(G) = x^a \text{Z}(G). Thus, \alpha = x^a z_1 for z_1 \in \text{Z}(G). Likewise \beta = x^b z_2.
    But then \alpha \beta = x^a z_1 x^b z_2 = x^{a+b} z_1 z_2 = x^{b} z_2 x^a z_1 = \beta \alpha.
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