Closure-characteristic subgroup
From Groupprops
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
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Definition
QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic
Symbol-free definition
A subgroup of a group is termed closure-characteristic if its normal closure in the whole group is a characteristic subgroup.
Definition with symbols
A subgroup H of a group G is termed closure-characteristic if the normal closure HG of H in G is characteristic in G.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Characteristic subgroup | invariant under all automorphisms | click here | ||
| Automorph-conjugate subgroup | all automorphic subgroups are conjugate | |||
| Join of automorph-conjugate subgroups | join of automorph-conjugate subgroups | |||
| Sylow subgroup | p-subgroup of finite group with index relatively prime to p | click here | ||
| Hall subgroup | subgroup of finite group whose order and index are relatively prime | |||
| Contranormal subgroup | normal closure is whole group |
Conjunction with other properties
Any normal subgroup that is also closure-characteristic, is characteristic.
Metaproperties
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
Join-closedness
YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: |
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness
Facts about Closure-characteristic subgroupRDF feed
| Defining ingredient | Normal closure +, and Characteristic subgroup + |
| Page class | Term + |
| Quick phrase | normal closure is characteristic +, and join of all conjugates is characteristic + |
| Satisfies metaproperty | Trim subgroup property +, Trivially true subgroup property +, Identity-true subgroup property +, Left-realized subgroup property +, Right-realized subgroup property +, and Join-closed subgroup property + |
| Weaker than | Characteristic subgroup +, Automorph-conjugate subgroup +, Join of automorph-conjugate subgroups +, Sylow subgroup +, Hall subgroup +, and Contranormal subgroup + |
