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:

$\displaystyle N_G(N_G(P)) = N_G(P) $. Show that equation isn't true for some subgroup H (not sylow) of $\displaystyle S_4 $ (symmetric group), meaning $\displaystyle N_G(N_G(H)) \neq N_G(H) $.

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 $\displaystyle S_4 $, created by the cycle (1234).

(a) Show that $\displaystyle C_G(H)=H $ for $\displaystyle C_G(H)= \{g \in G \| gh=hg \forall h \in H \} $

(b) Show that $\displaystyle N_G(H) $ is a 2-sylow subgroup of G.

I have some difficulty by showing that too.

I will very appreciate any help.