Convex subgroup of a partially ordered group

A subgroup of a -group such that for all ,

Many properties of po-groups can be described in terms of convex subgroups. If is convex subgroup of a po-group , then the set of right cosets of by is partially ordered with the induced order: if there exists an element such that .

The set of all convex subgroups of a totally ordered group (cf. -group) is well investigated. It is a complete chain, i.e., it is closed with respect to join and intersection. The system is infra-invariant, i.e., for all , . If for , , then . For an element there exist a maximal subgroup with the property , and a minimal subgroup with the property . The subgroup is normal in (cf. Normal subgroup) and the quotient group is order isomorphic to some subgroup of the additive group of real numbers. If is the normalizer of the subgroup (cf. also Normalizer of a subset), then and for every element , . Here, denotes the commutator subgroup of and . If, in a group , one can find a system with the properties listed above, then it is possible to turn into an -group such that is the system of convex subgroups for .

If is a locally nilpotent o-group, then the system is a central system of subgroups.

The set of all convex -subgroups (i.e., subgroups of that are sublattices of the lattice ) is very important for the description of the structure of an -group . It is a complete sublattice of the subgroup lattice of . If , then is isolated, i.e., . A subset of an -group is the kernel of an -homomorphism of if and only if ; in that case it is a normal subgroup of .

Also very important for describing the structure of an -group are the prime -subgroups, i.e., the convex -subgroups of such that the partially ordered set of right cosets is a chain. An -subgroup is prime if and only if it is convex and when for elements , . If , , and is maximal with respect to the property , then is prime. If , then is the intersection of a suitable set of prime subgroups. If , then there exists a natural -homomorphism from onto a transitive -subgroup of the -group , where is the group of order automorphisms of the totally ordered set of right cosets.

If in an -group , then the set

is called a polar and .

This article complements and updates the article Convex subgroup (in Volume 2).

References

[a1]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
[a2]	A. Bigard, K. Keimel, S. Wolfenstein, "Groupes et anneaux rétiqulés" , Springer (1977)
[a3]	"Lattice-ordered groups: advances and techniques" A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Kluwer Acad. Publ. (1989)
[a4]	V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)