# Subset-conjugacy-closed 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]

## Definition

### Definition with symbols

A subgroup $H$ of a group $G$ is termed subset-conjugacy-closed in $G$ if it satisfies the following equivalent conditions:

1. For any subsets $A,B$ of $H$, such that there exists $g \in G$ with $gAg^{-1} = B$, there exists $h \in H$ such that $hah^{-1} = gag^{-1}$ for all $a \in A$.
2. $H$ is a subset-conjugacy-determined subgroup of itself with respect to $G$, i.e., the fusion for subsets of $H$ in $G$, is contained in $H$.
3. $H$ possesses a distinguished set of coset representatives in $G$: In other words, there is a set $T$ of left coset representatives of $H$ in $G$ such that $hth^{-1} \in T$ for all $h \in H, t \in T$.

### Equivalence of definitions

Further information: Equivalence of definitions of subset-conjugacy-closed subgroup

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Central factor product with its centralizer equals whole group central factor implies subset-conjugacy-closed subset-conjugacy-closed not implies central factor |FULL LIST, MORE INFO
Retract has a normal complement; equivalently, image of a retraction retract implies subset-conjugacy-closed subset-conjugacy-closed not implies retract |FULL LIST, MORE INFO
Direct factor factor in an internal direct product (via retract, via central factor) (via retract, via central factor) Base of a wreath product, Central factor, Retract|FULL LIST, MORE INFO
Central subgroup contained in the center (via central factor) (via central factor) Central factor|FULL LIST, MORE INFO
Abelian abnormal subgroup abelian and an abnormal subgroup Abelian and abnormal implies subset-conjugacy-closed

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Weak subset-conjugacy-closed subgroup any two subsets conjugate in whole group are conjugate in subgroup; however, the conjugating element in the subgroup may not act the same element-wise
Conjugacy-closed subgroup any two elements of subgroup conjugate in whole group are conjugate in subgroup conjugacy-closed not implies subset-conjugacy-closed |FULL LIST, MORE INFO
Central factor of normalizer a central factor of its normalizer subset-conjugacy-closed implies central factor of normalizer central factor of normalizer not implies subset-conjugacy-closed |FULL LIST, MORE INFO
SCDIN-subgroup subset-conjugacy-determined subgroup in normalizer

## Metaproperties

### 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

### 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

### 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