let $A$ be a non empty set with a given structure, and let $G$ be the group of automorphisms of $A$, that is, the set of bijective maps from $A$ to itself that preserve its structure, with composition as group law.
Is it true that for any $a\in A$, the subgroup $G(a)$ of $G$ consisting of all elements of $G$ that preserve $a$ is a normal subgroup of $G$? Thanks in advance.