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.