So if we consider our group $N_G(A)$ and let the group act on the set A $\in$ P(G) via conjugation and consider the kernel we will get precisely the kernel being the centralizer, and since the kernel of an action is always a subgroup so the centralizer is a subgroup.
Similarly for the center we act on set G and we get the center. Now is it possible to get that the center is subgroup of the centralizer using only group actions ?