property of a normal subgroup of a transively group acting on a set $S$

Question

I'm trying to solve the part (c), but i can't. Please someone may give me a skill or key, i'll appreciate so much.

Answer 1

Suppose $y$ is some other point in $S$. By assumption, there exists some $h\in G$ such that $hx = y$. Now if $g\in G_x$, then $hgh^{-1}y = hgx = hx = y$, so $hgh^{-1}\in G_y$ (the converse holds as well). Use this property to show that if $N$ is normal and every element of $N$ fixes $x$, then every element of $N$ in fact fixes $y$ for every other $y\in S$.