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A classical result in group theory is that if all elements $g$ of a group $G$ satisfy $g^2=e$, then $G$ is an Abelian group.

My question is: Let $n\geq2$ be a positive integer. If all elements $g$ of a group $G$ satisfy $g^n=e$, for which values of $n$ is $G$ an Abelian group? For which values of $n$ can we provide an example of a non-Abelian group $G$?

There is an interesting result when $n=3$.

Kevin
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    Did you search the site? Probably we have covered many (if not all) cases alread. Anyway, the group of 3x3 uppertriangular matrices with 1s on the diagonal, and all entries from $\Bbb{Z}_n$ seems to work for all odd $n\ge 3$, and hence for multiples of such an $n$. That leaves powers of two, which is easy (the same set of matrices but entries from $\Bbb{Z}_2$. – Jyrki Lahtonen May 26 '24 at 10:02
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    Observe that if you can find examples whenever $n$ is an odd prime, and when $n=4$, you are done. – Jyrki Lahtonen May 26 '24 at 10:06
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    Dietrich found this. – Jyrki Lahtonen May 26 '24 at 10:08

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