Homomorph-dominating subgroup

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Definition

A subgroup $H$ of a group $G$ is termed homomorph-dominating in $G$ if, for any homomorphism $\phi \in \mathrm{H}\mathrm{o}\mathrm{m}(H,G)$, there exists $g\in G$ such that $\phi (H)\le gH{g}^{-1}$.

Relation with other properties

Stronger properties

• Order-dominating subgroup
• Homomorph-containing subgroup
• Sylow subgroup

Weaker properties

• Endomorph-dominating subgroup
• Isomorph-conjugate subgroup if the whole group is a co-Hopfian group -- it is not isomorphic to any proper subgroup of itself.

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