Variety-containing subgroup

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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

Definition with symbols

A subgroup $H$ of a group $G$ is termed a variety-containing subgroup if there exists a subvariety $\mathcal{V}$ of the variety of groups such that:

• $H\in \mathcal{V}$.
• If $K\le G$ is such that $K\in \mathcal{V}$, then $K\le H$.

Relation with other properties

For finite groups

Further information: Equivalence of definitions of variety-containing subgroup of finite group

In a finite group, the notion of variety-containing subgroup is equivalent to the notions of subhomomorph-containing subgroup and subisomorph-containing subgroup.

Stronger properties

• Normal Sylow subgroup
• Normal Hall subgroup

Weaker properties

• Homomorph-containing subgroup
• Subhomomorph-containing subgroup
• Subisomorph-containing subgroup
• Transfer-closed fully invariant subgroup
• Intermediately fully invariant subgroup
• Fully invariant subgroup
• Strictly characteristic subgroup
• Transfer-closed characteristic subgroup
• Intermediately characteristic subgroup
• Characteristic subgroup
• Homomorph-dominating subgroup
• Isomorph-containing subgroup
• Isomorph-dominating subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition