# Lattice-complemented subgroup

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]

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Lattice-complemented subgroup, all facts related to Lattice-complemented subgroup) |Survey articles about this | Survey articles about definitions built on this

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View a complete list of semi-basic definitions on this wiki

## Definition

### Symbol-free definition

A subgroup of a group is said to be **lattice-complemented** if there is another subgroup such that:

### Definition with symbols

A subgroup of a group is said to be **lattice-complemented** if there is another subgroup such that:

- is trivial

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups

View other monadic second-order subgroup properties

## Relation with other properties

### Stronger properties

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

A lattice-complemented subgroup of a lattice-complemented subgroup need not be lattice-complemented. `Further information: Lattice-complemented is not 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).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### Intermediate subgroup condition

NO:This subgroup property doesnotsatisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

If is a lattice-complemented subgroup of a group , and , is not necessarily lattice-complemented in . `For full proof, refer: Lattice-complemented does not satisfy intermediate subgroup condition`