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Thread: cyclic

  1. #1
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    cyclic

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

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