This is one part of a homework question. If we show this fact, then the rest of the problem is solved.
Let $G$ be a finite group and let $H$ be the subgroup generated by all Sylow p-subgroups. We want to show $H\lhd G$. Here is my reasoning so far:
At first, I thought that $H$ contained all of and only the elements of $G$ of order $p^k$ for some $k>0$. If this were the case, then $ghg^{-1}$ would have the same order as $h$, that is, $p^k$ for some $k$. That would mean that $ghg^{-1}\in H$ so we would be done if we could show that $H$ is not a p-group. For, if it were a p-group, it would contain a Sylow p-subgroup of $G$, a contradiction.
Also, my first statement about $H$ containing all of and only the elements of $G$ is suspicious. Any help would be appreciated. Thank you.