# Double coset-separated subgroup

From Groupprops

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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is termed **double coset-separated** if whenever two left cosets of the subgroup are conjugate (that is, there is an inner automorphism of the group mapping one bijectively to the other) then the two left cosets must lie in the same double coset of the subgroup.

### Definition with symbols

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## Relation with other properties

### Stronger properties

- Self-normalizing subgroup
- Abelian-quotient subgroup: In fact, this is the conjunction of being normal and double coset-separated
- Weakly cocentral subgroup
- Subgroup of double coset index two

### Weaker properties

## Metaproperties

### 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

From the definition, it is clear that if is double coset-separated in , and is any intermediate subgroup, then is also double coset-separated in .