Linearly ordered groups whose system of convex subgroups is central

Abstract

The order P on a group G is called rigid if for p e P the relation p¦[x, p]¦ɛ e P holds for every x e G, ɛ=±1. In this note we give criteria for the existence of a rigid linear order, for the extendability of a rigid partial order to a rigid linear order, and for the extendability of each rigid partial order to a rigid linear order on a group. It is proved that the class of groups each of whose rigid partial orders can be extended to a rigid linear order is closed with respect to direct products. A new proof of the theorem of M. I. Kargapolov which states that if a group G can be approximated by finite p-groups for infinite number of primes p, then it has a central system of subgroups with torsion-free factors is presented.