Convex subgroup of a partially ordered group
A subgroup of a
-group
such that for all
,
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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
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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) |
Convex subgroup of a partially ordered group. V.M. Kopytov (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_subgroup_of_a_partially_ordered_group&oldid=15705