C-closed self-centralizing subgroup

Symbol-free definition
A subgroup of a group is termed a c-closed self-centralizing subgroup or a centralizer of Abelian subgroup if it satisfies the following equivalent conditions:


 * 1) Its defining ingredient::centralizer equals its center and it equals the centralizer of its center (all relative to the whole group).
 * 2) It is self-centralizing (i.e., it contains its own centralizer in the whole group) and also occurs as the centralizer of some subgroup of the whole group.
 * 3) It occurs as the centralizer of some Abelian subgroup of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a c-closed self-centralizing subgroup or a centralizer of Abelian subgroup if it satisfies the following equivalent conditions:


 * 1) $$C_G(H) = Z(H)$$ and $$C_G(Z(H)) = H$$.
 * 2) $$C_G(H) \le H$$ and there exists a subgroup $$K \le G$$ such that $$H = C_G(K)$$.
 * 3) There exists an Abelian subgroup $$K$$ of $$G$$ such that $$H = C_G(K)$$.

Stronger properties

 * Weaker than::c-closed critical subgroup
 * Weaker than::Maximal among abelian subgroups

Weaker properties

 * Stronger than::c-closed subgroup
 * Stronger than::Self-centralizing subgroup