Self-centralizing normal subgroup

Symbol-free definition
A subgroup of a group is termed a self-centralizing normal subgroup or normal self-centralizing subgroup if it satisfies the following equivalent conditions:
 * It is normal and self-centralizing in the whole group (i.e., it contains its centralizer in the whole group).
 * The induced map from the quotient of the whole group by its center, to the automorphism group of the subgroup, is injective.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a self-centralizing normal subgroup if it satisfies the following equivalent conditions:
 * $$H \triangleleft G$$ and $$C_G(H) \le H$$.
 * The induced homomorphism $$G/Z(H) \to \operatorname{Aut}(H)$$, via the action by conjugation, is injective.

Stronger properties

 * Weaker than::Critical subgroup
 * Weaker than::Self-centralizing characteristic subgroup

Weaker properties

 * Stronger than::Coprime automorphism-faithful subgroup (for finite groups):
 * Stronger than::Normality-large normal subgroup and Stronger than::Normality-large subgroup: