# SCAB-subgroup

From Groupprops

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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]

## Definition

### Definition with symbols

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

- Any inner automorphism of restricts to a subgroup-conjugating automorphism of .
- Any subgroup-conjugating automorphism of restricts to a subgroup-conjugating automorphism of .
- is a normal subgroup of , and if are two subgroups of that are conjugate in , they are conjugate in .

## 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 Subgroup-conjugating automorphism.

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

Here is a left tight function restriction expression, showing that the property is a balanced subgroup property (function restriction formalism):

Subgroup-conjugating automorphism Subgroup-conjugating automorphism.

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

## Relation with other properties

### Stronger properties

### 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 transitiveABOUT 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition