Endomorph-dominating subgroup

From Groupprops

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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.
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RANDOM TIP:The relation with other properties section lists stronger and weaker properties, along with links to proofs of the implications and non-implications. This helps give a feel of how the subgroup property relates to other properties.

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 $\varphi(H) \le gHg^{-1}$ .