Polynomial-bound join-transitively subnormal 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]
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 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
- 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