Isomorph-dominating subgroup

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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, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]


BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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 \le 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