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Statement: Let G be a group and p a prime that divides $|G|$. Prove that if $K\le G$ such that $|K|$ is a power of p, K is contained in at least one Sylow p-group.

I just started studying Sylow p-groups, so although I'm familiar with Sylow theorems and a couple of corollaries, I don't know how to get started with this problem. Any hint is more than welcome.

PS: I looked for something related here at Math.SE but didn't find anything. Sorry if it's a duplicate.

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    Hint: Let $P$ be a Slow $p$-subgroup of $G.$ Consider the action of $K$ in the permutation action of $G$ on the right cosets of $P.$2012-05-27

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