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]

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If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed polynomial-bound join-transitively subnormal in $G$ if there exists a polynomial $f$ with integer coefficients such that if $K$ is a $k$-subnormal subgroup of $G$, $\langle H,K\rangle$ (the join of subgroups) is a $f(k)$-subnormal subgroup of $G$.

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

Relation with other properties

Stronger properties

• Normal subgroup: The polynomial is $f(k)=k$. For full proof, refer: Join of subnormal and normal implies subnormal of same depth
• 2-subnormal subgroup: The polynomial is $f(k)=2k$. 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

Weaker properties

• Join-transitively subnormal subgroup