SCAB-subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

Definition with symbols

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

1. Any inner automorphism of $G$ restricts to a 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

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

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 balanced subgroup property (function restriction formalism):

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 $H$.

Relation with other properties

Stronger properties

• Central factor
  • Central subgroup
  • Cocentral subgroup
  • Direct factor
  • Complemented central factor
• Hereditarily normal subgroup

Weaker properties

• Transitively normal subgroup
• Normal subgroup
• Right-transitively pronormal subgroup
• Right-transitively paranormal subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition