Endomorph-dominating subgroup

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Definition

A subgroup $H$ of a group $G$ is termed endomorph-dominating if for any endomorphism $\varphi$ of $G$, there exists $g\in G$ such that $\varphi (H)\leq gHg^{-1}$.

Relation with other properties

Stronger properties

• Order-dominating subgroup
• Homomorph-dominating subgroup

Weaker properties

• Automorph-conjugate subgroup, if the subgroup is a co-Hopfian group (i.e., it is not isomorphic to any proper subgroup of itself).