Originally Posted by

**aabsdr** Since S is a Sylow-p subgroup of G and K intersect S is a subgroup of S, we see that K intersect S is a p-subgroup of K.

Is that because K intersect S has to be either the identity e or S?

It remains to show that K intersect S is a maximal p-subgroup of K, which implies that K intersect S is a Sylow p-subgroup of K. This can be shown by using that [K:K intersect S] is coprime to p.

I am not seeing how we know that [K:K intersect S] is coprime to p.

We know that [G:S] is coprime to p.

what if |G|=p^2, then [G:S] would not be coprime to p right?