If $A$ is a normal subgroup of $G$, then $A ⊴ C(A)$ where $C(A)$ is the centralizer of $A$.
I am a bit confused. Here's my argument.
Given that $A$ is a normal subgroup of $G$. Thus, $gag^{-1} \in A $ for all $g \in G$ ------$1$
If $A$ is a normal subgroup of $C(A)$ then it must be the case $gag^{-1} \in A $ for all $g \in C(A)$
Pick $g \in C(A) $ then we have $gag^{-1} = a$ for all $a \in A$. since $C(A)$ is a subset of $G$ by $1$ it should follow $gag^{-1} \in A $ for all $g \in C(A)$ as well.
What I am missing here? I am not confident about the proof