# Join of homomorph-dominating subgroups

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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

A subgroup $H$ of a group $G$ is termed a join of homomorph-dominating subgroups if there exists a collection $H_i, i \in I$ of subgroups of $G$ such that $H$ is the join $\langle H_i \rangle_{i \in I}$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
homomorph-dominating subgroup every homomorphic image is contained in a conjugate (obvious) homomorph-domination is not finite-join-closed |FULL LIST, MORE INFO
homomorph-containing subgroup contains every homomorphic image (via homomorph-dominating) |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup whose normal closure is homomorph-containing normal closure is a homomorph-containing subgroup |FULL LIST, MORE INFO

## Formalisms

### In terms of the join-closure operator

This property is obtained by applying the join-closure operator to the property: homomorph-dominating subgroup
View other properties obtained by applying the join-closure operator