The Group Properties Wiki (pre-alpha)

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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.

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Conjunction with other properties 3 Metaproperties 3.1 Trimness 3.2 Join-closedness

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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 H^G 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