# 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 |