SCAB-subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a SCAB-subgroup if it satisfies the following conditions:


 * 1) Any defining ingredient::inner automorphism of $$G$$ restricts to a defining ingredient::subgroup-conjugating automorphism of $$H$$.
 * 2) Any subgroup-conjugating automorphism of $$G$$ restricts to a subgroup-conjugating automorphism of $$H$$.
 * 3) $$H$$ is a normal subgroup of $$G$$, and if $$A,B$$ are two subgroups of $$H$$ that are conjugate in $$G$$, they are conjugate in $$H$$.

Formalisms
The property of being a SCAB-subgroup has the following function restriction expression:

Inner automorphism $$\to$$ Subgroup-conjugating automorphism.

In other words, $$H$$ is a SCAB-subgroup of $$G$$ if and only if every inner automorphism of $$G$$ restricts to a subgroup-conjugating automorphism of $$H$$.

Here is a left tight function restriction expression, showing that the property is a :

Subgroup-conjugating automorphism $$\to$$ Subgroup-conjugating automorphism.

In other words, $$H$$ is a SCAB-subgroup of $$G$$ if and only if every subgroup-conjugating automorphism of $$G$$ restricts to a subgroup-conjugating automorphism of $$H$$.

Stronger properties

 * Weaker than::Central factor
 * Weaker than::Central subgroup
 * Weaker than::Cocentral subgroup
 * Weaker than::Direct factor
 * Weaker than::Complemented central factor
 * Weaker than::Hereditarily normal subgroup

Weaker properties

 * Stronger than::Transitively normal subgroup
 * Stronger than::Normal subgroup
 * Stronger than::Right-transitively pronormal subgroup
 * Stronger than::Right-transitively paranormal subgroup