Recently I noticed that if $G$ is a finite group and $g \in G$ for which the centralizer $C_G(g)$ is a normal subgroup, all of the elements of the conjugacy class $g^G$ commute with each other, and hence their product is a element of the center $Z(G)$ of $G$.
Now suppose that all of the centralizers of elements of $G$ are normal. Have these groups been classified? What can be said about these groups? I noticed that if $P$ is any Sylow $p$-subgroup of $G$ and $z \in Z(P)$, then $G=N_G(P)C_G(z)$ by the Frattini argument.