# Subset-conjugacy-closed 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 **subset-conjugacy-closed** in if it satisfies the following equivalent conditions:

- For any subsets of , such that there exists with , there exists such that for all .
- is a subset-conjugacy-determined subgroup of itself with respect to , i.e., the fusion for subsets of in , is contained in .
- possesses a distinguished set of coset representatives in : In other words, there is a set of left coset representatives of in such that for all .

### 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 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

### 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