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.
Best Answer
I believe the comment by James is correct, these groups are precisely the $2$-Engel groups.
Claim: The following statements are equivalent for a group $G$.
Every centralizer in $G$ is a normal subgroup.
Any two conjugate elements in $G$ commute, ie. $x^g x = x x^g$ for all $x, g \in G$.
$G$ is a $2$-Engel group, ie. $[[x,g],g] = 1$ for all $x, g \in G$.
Proof:
1) implies 2): $x \in C_G(x)$, thus $x^g \in C_G(x)$ since $C_G(x)$ is normal.
2) implies 3): $x^g = x[x,g]$ commutes with $x$, thus $[x,g]$ also commutes with $x$.
3) implies 1): If $[[x,g],g] = 1$ for all $g \in G$, then according to Lemma 2.2 in [*], we have $[x, [g,h]] = [[x,g],h]^2$. Therefore $[C_G(x), G] \leq C_G(x)$, which means that $C_G(x)$ is a normal subgroup.
[*] Wolfgang Kappe, Die $A$-Norm einer Gruppe, Illinois J. Math. Volume 5, Issue 2 (1961), 187-197. link