Thread: Group Theory - Sylow and Conjugation

A couple of questions I had hard time solving (and not succeeded).

1. We showed that for every p-sylow group P in G, the following is true:
. Show that equation isn't true for some subgroup H (not sylow) of (symmetric group), meaning .
I don't fully understand how to do this without doing it brute-force, by applying all elements of G on the chosen H.

2. H is a subgroup of , created by the cycle (1234).
(a) Show that for
(b) Show that is a 2-sylow subgroup of G.

I have some difficulty by showing that too.

I will very appreciate any help.

Originally Posted by pavelr

1. We showed that for every p-sylow group P in G, the following is true:
. Show that equation isn't true for some subgroup H (not sylow) of (symmetric group), meaning .
I don't fully understand how to do this without doing it brute-force, by applying all elements of G on the chosen H.

I did not actually work out the details for this problem, but I had an idea that you might want to try. Let i.e. all permutations that fix 4. Then it is seem that (in fact is naturally isomorphic to ). Now since is not normal it means . If i.e. if there is so that then it means is propely contained in . By Lagrange's theorem it would mean since . But . Therefore, . I cannot gaurentte you this would work because I was kinda lazy to work out the details but this would be the first thing I would try to do.