Automorphically endomorph-dominating subgroup

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]

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Definition

A subgroup $H$ of a group $G$ is termed automorphically endomorph-dominating if, for any endomorphism $\sigma$ of $G$, there exists an automorphism $\varphi$ of $G$ such that $\sigma (H)\leq \varphi (H)$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
endomorph-dominating subgroup	image under any endomorphism is contained in one of its conjugate subgroups	follows from the fact that conjugate subgroups are automorphic subgroups		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subgroup whose characteristic closure is fully invariant	its characteristic closure is a fully invariant subgroup			|FULL LIST, MORE INFO
subgroup whose characteristic core is fully invariant	its characteristic core is a fully invariant subgroup			|FULL LIST, MORE INFO