Let $G$ be a group and $$N\unlhd G$$ Suppose that $G$ acts transitively on a set $\Omega$ and that $N$ fixes some $x \in\Omega$, namely $n\cdot x = x$ for all $n \in N$.
How are $N$ and the stabilizer $Gx$ related?
My thoughts: - I have seen somewhere that 'if the group action is transitive, then the stabilizer is normal' and that a 'normal subgroup normalizes a stabilizer' but I'm not sure if these statements are true. If they are could I, first of all, show that the stabilizer is a subgroup then go on to show it is a normal subgroup? Another thought I had was about cardinality so since $G$ is transitive if $N$ is normal then all orbits of $N$ on $X$ have the same cardinality...
I may be completely wrong so any tips or hints would be very appreciated thank you.
The two statements you quote are false. For example $G=S_n$ acts transitively but the stabiliser of a point $G_x$ is not normal in $G$, also $A_n$ is normal in $S_n$, but does not normalise $G_x$.
A subgroup $H$ of $G$ fixes $x$ if and only if $H\le G_x$.