Subgroup for which every lattice complement is a permutable complement

From Groupprops
Jump to: navigation, search
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]

Definition

A subgroup of a group is termed a subgroup for which every lattice complement is a permutable complement if it has the property that any lattice complement to it is a permutable complement to it.

In particular, if such a subgroup is a lattice-complemented subgroup (i.e., it has at least one lattice complement) then it is a permutably complemented subgroup.

Relation with other properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subgroup |FULL LIST, MORE INFO
Permutable subgroup permutes with every subgroup |FULL LIST, MORE INFO
Subgroup that permutes with every subgroup intersecting it trivially