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]


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

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