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 $G$ and $L$ is a subgroup of $H$ 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

Variety-containing subgroup
Subhomomorph-containing subgroup: For proof of the implication, refer subhomomorph-containing implies subisomorph-containing and for proof of its strictness (i.e. the reverse implication being false) refer subisomorph-containing not implies subhomomorph-containing.

Weaker properties