Lattice-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]

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 definition
VIEW: Definitions built on this | Facts about this: (facts closely related 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


Symbol-free definition

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

  • The two subgroups intersect trivially
  • The join of the two subgroups is the whole group

Definition with symbols

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

  • H\cap K is trivial
  • \langle H,K \rangle = G


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



NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT 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


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 does not satisfy 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 condition
ABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition

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