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o-Automorphisms of o-Groups of Finite Rank
by
Ramiro H. Lafuente-Rodriguez
Bowling Green State University
A group G is divisible if for every g in G and every natural number n, there exists x in G such that xn=g. The classical unanswered question in the theory of totally ordered groups (o-groups) is whether or not every o-group can be embedded in a divisible o-group. W.C.Holland proved in 1961 the existence of an example of an o-group of archimedean rank 3 that is not embeddable in a divisible o-group of archimedean rank 3. This example uses the non-divisible o-group o-Aut(G), where G=RxR (ordered lexicographically). We will discuss a more general case, when the o-group G is an o-extension of R by R, and prove that o-Aut(G) is divisible when the extension is not central. We will also discuss the case when G is of rank greater than 2, and prove the following
Theorem: Let G be the semidirect product Rn× < a > , where < a > is an infinite cyclic group and a induces an action on Rn with no square root. G can be embedded in an o-group of archimedean rank at most 2n where a has a square root.
Date received: January 31, 2002
Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cain-13.