Conjugate-commensurable 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

Symbol-free definition

A subgroup of a group is termed conjugate-commensurable if it is commensurable with each of its conjugate subgroups. Equivalently, its commensurator in the whole group is the whole group.

Definition with symbols

A subgroup H of a group G is termed a conjugate-commensurable subgroup if, for any g \in G, H \cap gHg^{-1} has finite index in both H and gHg^{-1}.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup Nearly normal subgroup|FULL LIST, MORE INFO
subgroup of finite group
finite subgroup
subgroup of finite index Automorph-commensurable subgroup, Nearly normal subgroup|FULL LIST, MORE INFO
nearly normal subgroup subgroup of finite index in its normal closure
isomorph-commensurable subgroup commensurable with every isomorphic subgroup
automorph-commensurable subgroup commensurable with every automorphic subgroup