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Is there any non-abelian group of order $n=49$? I assume there should be at least one but I cannot find an example.

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    $\begingroup$ Actually any group of order $p^2$ for $p$ a prime is abelian. This is a nice exercise using the class equation. $\endgroup$ – Qiaochu Yuan Feb 25 '12 at 0:17
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    $\begingroup$ $49 = 7^2$. For a given prime $p$ there are only two groups of order $p^2$, and they are both abelian. You should try proving this! $\endgroup$ – Dylan Moreland Feb 25 '12 at 0:18
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    $\begingroup$ I took the liberty of removing the boldface text... that seems completely unnecessary. $\endgroup$ – Tyler Feb 25 '12 at 2:39
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The only groups of order $49$ are $\mathbb Z_{49}$ and $\mathbb Z_7\times \mathbb Z_7$ since $49$ is a square of a prime number.

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  • $\begingroup$ both are cyclic though right? $\endgroup$ – cele Oct 29 '19 at 22:59

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