# Sub-homomorph-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]

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with subhomomorph-containing subgroup

## Definition

A subgroup of a group is termed a **sub-homomorph-containing subgroup** if there is a chain of subgroups:

such that each is a homomorph-containing subgroup of .

## Formalisms

### In terms of the subordination operator

This property is obtained by applying the subordination operator to the property: homomorph-containing subgroup

View other properties obtained by applying the subordination operator

## Relation with other properties

### Stronger properties

- Homomorph-containing subgroup
- Subhomomorph-containing subgroup
- Normal Sylow subgroup
- Normal Hall subgroup

### Weaker properties

- Fully invariant subgroup
- Strictly characteristic subgroup
- Sub-isomorph-containing subgroup
- Characteristic subgroup
- Normal 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

### 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