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tyrannosaurus
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Homework Statement
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?
Homework Equations
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