(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose K is a normal subgroup of a finite group G and S is a p-Sylow subgroup of G. Prove that K intersect S is a p-Sylow subgroup of K. So I know that K is a unique p-sylow group by definition, is that enough to prove that the intersection of K with S is a p-sylow subgroup of K?

2. Relevant equations

3. The attempt at a solution

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

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

THe hints I was given to finish this off was that [K: K intersect S] = [KS:S] and [G:S] = [G:KS][KS:S] and from that we are suppose to get that [K:K intersect S] is relatively prime to p.

I am not getting how we know that [K: K intersect S] = [KS:S] and that [G:S] = [G:KS][KS:S] and how this implies that [K: K intersect S] is relatively prime to p.

I know that KS is a subgroup of G since K is a normal subgroup of G. If anyone could help me with this problem I will be entirnaly gratefull

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: P-Sylow subgroup question

**Physics Forums | Science Articles, Homework Help, Discussion**