Join of homomorph-dominating subgroups

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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
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)	|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