Conjugate-commensurable subgroup: Difference between revisions

Revision as of 21:00, 26 May 2010

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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	Intermediate notions
Normal subgroup		\|FULL LIST, MORE INFO
Subgroup of finite group
Finite subgroup
Subgroup of finite index		\|FULL LIST, MORE INFO
Nearly normal subgroup
Isomorph-commensurable subgroup	commensurable with every isomorphic subgroup
Automorph-commensurable subgroup	commensurable with every automorphic subgroup

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 ===Symbol-free definition===
-A subgroup of a group is termed '''conjugate-commensurable''' if it is [[defining ingredient::commensurable subgroups|commensurable]] with each of its [[defining ingredient::conjugate subgroups]].
+A subgroup of a group is termed '''conjugate-commensurable''' if it is [[defining ingredient::commensurable subgroups|commensurable]] with each of its [[defining ingredient::conjugate subgroups]]. Equivalently, its [[defining ingredient::commensurator of a subgroup|commensurator]] in the whole group is the whole group.
 ===Definition with symbols===