Finite groups of which the centralizer of each element is normal. 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.

The first observation is that such a finite group is nilpotent. For, given any $x \in G,$ we certainly have $x \in F(C_{G}(x)),$ and the latter group is contained in $F(G)$ as $C_{G}(x) \lhd G.$ This essentially reduces the question to one about $p$-groups. Notice that any finite $p$-group $P$ of nilpotency class $2$ has the property (thanks to Jack Schmidt for pointing out an earlier error). I am not sure at the moment whether it is possible to characterize $p$-groups with the property in any straightforward fashion.
