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Suppose that $n \gt m \ge 3$. Prove that if $m$ does not divide $n$, then $D_m$ is not isomorphic to any subgroup of $D_n$.

I know that since isomorphisms preserve order, if there has to exist an isomorphism between $D_m$ and $D_n$, m has to divide n other wise there would exist an element in $D_n$ that cannot be mapped to any element in $D_m$, but im not sure exactly how to show this.

Any input would be appreciated,


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up vote 3 down vote accepted

By Lagrange's theorem if $D_m$ was isomorphic to a subgroup of $D_n$, $|D_m|=2m$ would need to divide $|D_n|=2n$. Can you spot a contradiction with your hypothesis?

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(+1) Straightforward way. – Babak S. Feb 24 '13 at 19:54

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