Is there $G$ subgroup of $S_n$ that is abelian but not cyclic?
I found: $G=\left\{id,\left(2\:3\right)\left(1\:4\right),\left(2\:4\right)\left(1\:3\right),\left(3\:4\right)\left(1\:2\right)\right\}$
It is right? And there is maybe an easier example?
2026-04-06 03:08:18.1775444898
Abelian subgroup but not cyclic
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It's right. You found the copy of the Klein four subgroup in $S_4$ which is normal. There are $3$ others which are not normal.
Since $S_4\hookrightarrow S_{4+k} $, this example will "work" for any $n\ge4$ (but without the normality property ).
You could also use Cayley's theorem to embed any finite, noncyclic abelian group in some $S_n $.