The Group Properties Wiki (pre-alpha)
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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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RANDOM SUBGROUP PROPERTY: Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.
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Definition
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
- Characteristic subgroup
- Automorph-conjugate subgroup
- Join of automorph-conjugate subgroups
- Sylow subgroup
- Hall subgroup
- Contranormal subgroup
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 all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties
Join-closedness
This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property
View a complete list of join-closed subgroup properties
Categories: Subgroup properties | Terminology local to the wiki | Nonstandard terminology | Normal-to-characteristic subgroup properties | Trim subgroup properties | Trivially true subgroup properties | Identity-true subgroup properties | Left-realized subgroup properties | Right-realized subgroup properties | Join-closed subgroup properties

