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Groupprops β

Homomorph-dominating subgroup

Definition

A subgroup H of a group G is termed homomorph-dominating in G if, for any homomorphism \varphi \in \operatorname{Hom}(H,G), there exists g \in G such that \varphi(H) \le gHg^{-1}.

Relation with other properties

Stronger properties

Weaker properties

Conjunction with other properties

A homomorph-containing subgroup is precisely the same as a subgroup that is both normal and homomorph-dominating. For full proof, refer: Homomorph-dominating and normal equals homomorph-containing