Suppose I have a group $G$ which acts on a set $X$. Suppose I can find $H$ a subgroup of $G$ which acts trivially on $X$. Does it then automatically mean that the action factors through?
I can see it is we can show that $H$ is a normal subgroup of $G$... If $H$ acts trivially on $X$ then is it always a normal subgroup of $G$? Thank you.