# Alternate proof of Sylow's First theorem.

I am trying to prove first Sylow Theorem using the Lemma: if $$G$$ is a finite group such that has a Sylow $$p$$-subgroup and $$H\subset G$$, then $$H$$ has a Sylow $$p$$-subgroup.

The way I want to go about proving first Sylow Theorem using lemma is to notice that any finite group $$G$$ injects into $$S_{|G|}$$.

Given the Lemma, it suffices to show that $$S_{|G|}$$ has a Sylow $$p$$-subgroup. If $$p^k$$ is the highest power of $$p$$ such that $$p^k| |G|$$, then I can easily see that $$\langle(123\cdots p^k)\rangle\subset S_{|G|}$$ is a subgroup with order $$p^k$$. But this doesn't finish the proof as $$|G|!$$ may be divisible by $$p^{k+1}$$, which then means this subgroup is not a Sylow $$p$$-subgroup of $$S_{|G|}$$.

Would there be hints about what steps I should make in this way of proving the first Sylow Theorem?

• Thanks for your comment, but I was more seeking a proof of the fact that every $S_n$ has a $p-$Sylow subgroup. Because if I can show this, then I get an alternative proof of first Sylow theorem other than shown in the post that you suggeste Mar 14, 2022 at 21:24
• You should like divide your steps into a list , like 1. , 2. , 3... So it becomes easier to understand but that is just my opinion. I've done some editing to make it clear to someone else that you require help on alternate method Mar 14, 2022 at 21:34
• Serre prefers to do this by embedding $H$ not into $S_n$ where $n = |H|$, but instead into ${\rm GL}_n(\mathbf F_p)$ by having $H$ act as permutations of the basis of $\mathbf F_p^n$ (index basis by elements of $H$), since it is much cleaner to describe a $p$-Sylow subgroup of ${\rm GL}_n(\mathbf F_p)$ then a $p$-Sylow subgroup of $S_n$.
– KCd
Mar 14, 2022 at 22:37
• I see that indeed resolves my question thanks. Mar 14, 2022 at 23:11
• If you want to know more about the Sylow subgroups of symmetric groups, then I recommed Ted's explanation. You may want to learn about wreath products, but the case $p=2$ is simpler, and the generalization to other primes is not too difficult either. For example, the permutations $(123)$ and $(147)(258)(369)$ generate a Sylow $3$-subgroup of $S_9$ etc. Mar 15, 2022 at 4:43