Conjugate-commensurable subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A subgroup of a group is termed '''conjugate-commensurable''' if it is [[defining ingredient::commensurable subgroups|commensurable]] with | 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=== | ===Definition with symbols=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Weaker than:: | | [[Weaker than::normal subgroup]] || || || || {{intermediate notions short|conjugate-commensurable subgroup|normal subgroup}} | ||
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| [[Weaker than:: | | [[Weaker than::subgroup of finite group]] || || || || | ||
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| [[Weaker than:: | | [[Weaker than::finite subgroup]] || || || || | ||
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| [[Weaker than:: | | [[Weaker than::subgroup of finite index]] || || || || {{intermediate notions short|conjugate-commensurable subgroup|subgroup of finite index}} | ||
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| [[Weaker than:: | | [[Weaker than::nearly normal subgroup]] || [[subgroup of finite index]] in its [[normal closure]] || || || | ||
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| [[Weaker than:: | | [[Weaker than::isomorph-commensurable subgroup]] || commensurable with every isomorphic subgroup || || || | ||
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| [[Weaker than:: | | [[Weaker than::automorph-commensurable subgroup]] || commensurable with every [[automorphic subgroups|automorphic subgroup]] || || || | ||
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Latest revision as of 15:04, 9 March 2020
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
This is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup
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 of a group is termed a conjugate-commensurable subgroup if, for any , has finite index in both and .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | 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 | subgroup of finite index in its normal closure | |||
| isomorph-commensurable subgroup | commensurable with every isomorphic subgroup | |||
| automorph-commensurable subgroup | commensurable with every automorphic subgroup |