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How we can show that {1} and {1,-1} are the only finite groups of nonzero real numbers under binary operations multiplication?

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The only torsion elements (equivalently, roots of unity) in ${\bf R}^\times$ are $\pm1$. –  anon Mar 12 '13 at 5:17
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Suppose that $|a|\ne 1$, $a\ne 0$. Show that if $m\ne n$, then $a^m \ne a^n$.

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