Subisomorph-containing subgroup
From Groupprops
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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]
If the ambient group is a finite group, this property is equivalent to the property: variety-containing subgroup
View other properties finitarily equivalent to variety-containing subgroup | View other variations of variety-containing subgroup |
Definition
A subgroup of a group
is termed subisomorph-containing if whenever
is a subgroup of
and
is a subgroup of
such that
and
are isomorphic, then
is also a subgroup of
.
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 |