# Subgroup for which every lattice complement is a permutable complement

From Groupprops

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 |