TITLE: About stabilizer in group action
QUESTION [6 upvotes]: Let $X$ be a finite set and $x$ is an element of $X$. Let $G_x$, the stabilizer subgroup, be the subset of $S_X$ consisting of permutations that fix $x$.
The question is 

Is stabilizer always a normal subgroup?

I know that $G_x=\{g\in G\mid gx=x, x \in X\}$ and I have proved that stabilizer is a subgroup of $G$. But I got stuck at the point proving stabilizer is always normal.
I have found that 'if the group action is transitive, then the stabilizer is normal'.
So, I came to a conclusion that stabilizers are not always normal. But I cannot understand this. Is there some good explanation on this (stabilizer is not always normal) with examples?
Thank you.

REPLY [5 votes]: If this where true all subgroups would be normal. Take a subgroup $H$ of $G$. Consider the left regular action of $G$ on the set of left cosets of $G$ by $H$. The stabilizer of $H$ is clearly $H$. Using the above we conclude $H$ is normal.