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.
Best Answer
Take the action of $S_3$ on itself by conjugation and $x=(1\ 2)$. Then the stabilizer of $x$ is $\{e, x\}$ and this is not a normal subgroup of $S_3$.