# Linear-bound join-transitively subnormal subgroup

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

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:subnormal subgroup

View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

This is a variation of join-transitively subnormal subgroup|Find other variations of join-transitively subnormal subgroup |

## Definition

A subgroup of a group is termed **linear-bound join-transitively subnormal** if there exists a natural number such that, given any -subnormal subgroup of , the join is -subnormal.

Here, a -subnormal subgroup is a subgroup whose subnormal depth is at most .

## Relation with other properties

### Stronger properties

- Normal subgroup: For a normal subgroup, we can set .
`For full proof, refer: Join of normal and subnormal implies subnormal of same depth` - 2-subnormal subgroup: For a 2-subnormal subgroup, we can set .
`For full proof, refer: 2-subnormal implies join-transitively subnormal` - Subnormal subgroup of finite index
- Intermediately linear-bound join-transitively subnormal subgroup
- Linear-bound intermediately join-transitively subnormal subgroup
- Asymptotically fixed-depth join-transitively subnormal subgroup

### Weaker properties

- Polynomial-bound join-transitively subnormal subgroup
- Join-transitively subnormal subgroup
- Subnormal subgroup

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

This follows from the fact that normal subgroups are linear-bound join-transitively subnormal, but not all subnormal subgroups are linear-bound join-transitively subnormal. That in turn follows from the fact that subnormality is not finite-join-closed. `For full proof, refer: Join of two 3-subnormal subgroups need not be subnormal, subnormality is not finite-join-closed`

If are both linear-bound join-transitively subnormal with corresponding natural numbers , their join is linear-bound join-transitively subnormal with natural number . (A smaller natural number might also work for the join).