Subisomorph-containing 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]

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If the ambient group is a finite group, this property is equivalent to the property: variety-containing subgroup
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Definition

A subgroup $H$ of a group $G$ is termed subisomorph-containing if whenever $K$ is a subgroup of $H$ and $L$ is a subgroup of $G$ such that $K$ and $L$ are isomorphic, then $L$ is also a subgroup of $H$.

Relation with other properties

In groups with specific properties

• Finite group and periodic group: In a finite group and more generally in a periodic group, the notion of subisomorph-containing subgroup coincides with the notions of subhomomorph-containing subgroup and variety-containing subgroup. Further information: Equivalence of definitions of variety-containing subgroup of finite group, Equivalence of definitions of variety-containing subgroup of periodic group
• Group of prime power order: For groups of prime power order, subisomorph-containing subgroups must be omega subgroups of group of prime power order, though the converse, while true for regular p-groups, is not always true. For full proof, refer: Variety-containing implies omega subgroup in group of prime power order, omega subgroups are variety-containing in regular p-group, omega subgroups not are variety-containing

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
variety-containing subgroup	contains every subgroup of the whole group in the variety generated by it			Subhomomorph-containing subgroup|FULL LIST, MORE INFO
subhomomorph-containing subgroup	contains every subgroup of the whole group isomorphic to a subquotient of it			|FULL LIST, MORE INFO
normal Sylow subgroup	normal subgroup of prime power order whose order and index are relatively prime			Variety-containing subgroup|FULL LIST, MORE INFO
normal Hall subgroup	subgroup whose order and index are relatively prime			Variety-containing subgroup|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorph-containing subgroup	contains every isomorphic subgroup			Right-transitively isomorph-containing subgroup|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup	invariant under all injective endomorphisms			Isomorph-containing subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup	injective endomorphism-invariant in every intermediate subgroup			Isomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup	invariant under every automorphism			Intermediately characteristic subgroup, Isomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Transfer-closed characteristic subgroup|FULL LIST, MORE INFO
intermediately characteristic subgroup	characteristic in every intermediate subgroup			Isomorph-containing subgroup, Transfer-closed characteristic subgroup|FULL LIST, MORE INFO
transfer-closed characteristic subgroup	intersection with any subgroup is characteristic in that subgroup			|FULL LIST, MORE INFO