A p subgroup of a group is contained in a sylow p subgroup

Question

I am trying to solve a problem involving automorphism of a group.There needs the following argument: a p subgroup is contained in a sylow p subgroup.Is it true?I can't prove it,may be it is elementary.Plz help me. I tried it producting with a sylow p subgroup,although its order is p power bt it may not be a group since there is no normal argument is given.

Answer 1

If a Sylow $p$-subgroup $P$ of group $G$ is by definition a maximal $p$-subgroup (i.e. if $P\leq Q\leq G$ and $Q$ is a $p$-subgroup then $P=Q$) then we can prove it like this:

If $H$ is a $p$-subgroup then there are $2$ possibilities: $H$ is a maximal $p$-subgroup (then $H\leq H$ tells us that $H$ is contained in a maximal $p$-subgroup) or we can find a $p$-subgroup $H'$ that properly contains $H$. In the second case we can repeat that for $H'$ and if $G$ is finite then eventually we will arrive at a maximal $p$-subgroup that contains $H$. If $G$ is not finite then it can be shown by means of Zorn's lemma.