Double coset-separated subgroup

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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

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