# Join of homomorph-dominating subgroups

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]

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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
join of Sylow subgroups join of Sylow subgroups (with no restriction on whether the primes need be the same or different) of the whole group follows from Sylow implies homomorph-dominating |FULL LIST, MORE INFO
Hall subgroup order and index are relatively prime (via join of Sylow subgroups) (via join of Sylow subgroups) Join of Sylow subgroups|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