# Variety-containing 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]

## Definition

### Definition with symbols

A subgroup of a group is termed a **variety-containing subgroup** if there exists a subvariety of the variety of groups such that:

- .
- If is such that , then .

## 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

### 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.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### 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 conditionABOUT 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