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Let $G$ be a group and $1$ is the identity in $G$. Suppose $a$, $b$ in $G$ and $ab=1$, how could one simply show that $ba=1$?

Thanks!

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    Multiply $ab=1$ on the left by $a^{-1}$.2011-10-08
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    @Anonymous. Now can you prove this: if $z$ is an element of the center $Z(G)$ of $G$, and let $a, b$ in $G$ with $ab=z$. Then $ba=z$.2011-10-09

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