Strongly permutably complemented 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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Definition

A subgroup $H$ of a group $G$ is termed strongly permutably complemented in $G$ if there exists a permutable complement $K$ to $H$ in $G$, such that, for any subgroup $L\le H$, $K$ and $L$ are permuting subgroups.

Relation with other properties

Stronger properties

• Retract

Weaker properties

• Permutably complemented subgroup
• Lattice-complemented subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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If $H\le K\le G$ are such that $H$ is strongly permutably complemented in $K$ and $K$ is strongly permutably complemented in $G$, then $H$ is strongly permutably complemented in $G$. For full proof, refer: Strongly permutably complemented is transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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