Name of a subgroup whose centralizer is equal to the center of the group

Question

Let $G$ be a group. Is there a specific name for those subgroups $H<G$ such that the centralizer $C_G(H)<G$ is equal to the center $Z(G)<G$ ?

Answer 1

I couldn't find a name for this either. There is a weaker but related notion on groupprops, but it doesn't seem to have a special name: Subgroup whose center is contained in the center of the whole group

It can be stated equivalently as:

1. $Z(H) \leq Z(G)$
2. $Z(H) = Z(G) \cap H$
3. $C_G(H) \cap H = Z(G) \cap H$
4. $C_G(H) \cap H = Z(H)$

whereas OP's property asks (3) to hold in all of $G$.

Answer 2

For compact real Lie groups, there is a similar notion called "S-subgroups" by Dynkin. The original definition is $H\leq G$ is an S-subgroup if the only regular subgroup of $G$ containing $H$ is itself. This definition is equivalent to $C_{G}(H)=Z(G)$ (Proposition 4 of Minchenko's paper The Semisimple subalgebras of exceptional Lie algebras).

Since finite groups can be viewed as compact Lie groups, if you like you can call it an S-subgroup. But I think the original definition of S-subgroups is meaningless for finite groups.