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]

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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)	|FULL LIST, MORE INFO
Central subgroup	contained in the center	(via central factor)	(via 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