Automorph-commensurable subgroup
From Groupprops
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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 an automorph-commensurable subgroup if it is commensurable with all its automorphic subgroups.
Definition with symbols
A subgroup of a group
is termed an automorph-commensurable subgroup if, for any automorphism
of
,
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 |
---|---|---|---|---|
Characteristic subgroup | ||||
Finite subgroup | ||||
Subgroup of finite group | ||||
Subgroup of finite index | ||||
Isomorph-commensurable subgroup |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
Conjugate-commensurable subgroup |