# Polynomial-bound join-transitively subnormal 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: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

### Definition with symbols

A subgroup of a group is termed **polynomial-bound join-transitively subnormal** in if there exists a polynomial with integer coefficients such that if is a -subnormal subgroup of , (the join of subgroups) is a -subnormal subgroup of .

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

## Relation with other properties

### Stronger properties

- Normal subgroup: The polynomial is .
`For full proof, refer: Join of subnormal and normal implies subnormal of same depth` - 2-subnormal subgroup: The polynomial is .
`For full proof, refer: 2-subnormal implies join-transitively subnormal` - Linear-bound join-transitively subnormal subgroup: The polynomial is linear in this case.
- Subnormal-permutable subnormal subgroup:
`For full proof, refer: Subnormal-permutable and subnormal implies join-transitively subnormal`