If $G$ is a $p$-group such it's order is $p^n$, is it possible that some Sylow $p$-group $S$ from say $H$, is isomorphic to $G$? Under what conditions that will happen?
2026-04-06 23:01:43.1775516503
$p$-group isomorphic to some $p$-Sylow
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Take another prime $q \neq p$ and take $\mathbb{Z}_q$ make a new group H by direct product of G and $\mathbb{Z}_q$. G is a sylow p subgroup of H here. It will always happen.