Isomorph-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 an isomorph-dominating subgroup if, for any subgroup $K$ of $G$ such that $H$ and $K$ are isomorphic groups, $K$ is contained in a conjugate subgroup of $H$, i.e., there exists $g\in G$ such that $K\leq gHg^{-1}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
homomorph-dominating subgroup	every homomorphic image is contained in a conjugate			|FULL LIST, MORE INFO
isomorph-containing subgroup	contains every isomorphic subgroup			|FULL LIST, MORE INFO
homomorph-containing subgroup	contains every homomorphic image			|FULL LIST, MORE INFO
isomorph-free subgroup	no other isomorphic subgroup			|FULL LIST, MORE INFO
isomorph-conjugate subgroup	conjugate to every isomorphic subgroup			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
automorph-dominating subgroup	every automorph is contained in a conjugate			Template:Intermediate notionsshort