Ordered Groups and Infinite Permutation Groups PDF
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Ordered Groups and Infinite Permutation Groups Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science. Amsterdam. The Netherlands Volume 354 Ordered Groups and Infinite Permutation Groups edited by W. Charles Holland Bowling Green State University KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13: 978-1-4613-3445-3 e-ISBN-13: 978-1-4613-3443-9 001: 10.1007/978-1-4613-3443-9 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved © 1996 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Preface ...................................................................... vii V. M. KOPYTOV AND N. YA. MEDVEDEV Quasivarieties and Varieties of Lattice-Ordered Groups ......................... 1 S. H. MCCLEARY Lattice-ordered Permutation Groups: The Structure Theory .................... 29 J. K. TRUSS On Recovering Structures from Quotients of their Automorphism Groups ...... 63 M. DROSTE AND R. GOBEL The Automorphism Groups of Generalized McLain Groups ................... 97 M. RUBIN Locally Moving Groups and Reconstruction Problems ........................ 121 S. A. ADELEKE Infinite Jordan Permutation Groups ........................................ 159 C. E. PRAEGER The Separation Theorem for Group Actions ................................. 195 D. MACPHERSON Permutation Groups Whose Subgroups Have Just Finitely Many Orbits ...... 221 J. L. ALPERIN, J. COVINGTON, AND D. MACPHERSON Automorphisms of Quotients of Symmetric Groups .......................... 231 v Preface The subjects of ordered groups and of infinite permutation groups have long en joyed a symbiotic relationship. Although the two subjects come from very different sources, they have in certain ways come together, and each has derived considerable benefit from the other. My own personal contact with this interaction began in 1961. I had done Ph. D. work on sequence convergence in totally ordered groups under the direction of Paul Conrad. In the process, I had encountered "pseudo-convergent" sequences in an ordered group G, which are like Cauchy sequences, except that the differences be tween terms of large index approach not 0 but a convex subgroup G of G. If G is normal, then such sequences are conveniently described as Cauchy sequences in the quotient ordered group GIG. If G is not normal, of course GIG has no group structure, though it is still a totally ordered set. The best that can be said is that the elements of G permute GIG in an order-preserving fashion. In independent investigations around that time, both P. Conrad and P. Cohn had showed that a group admits a total right ordering if and only if the group is a group of automor phisms of a totally ordered set. (In a right ordered group, the order is required to be preserved by all right translations, unlike a (two-sided) ordered group, where both right and left translations must preserve the order.) It was at about that time that I had the good fortune to be able to attend the lectures of Helmut Wielandt and Olaf Tamaschke in Tiibingen. Wielandt's lecture notes on infinite permutation groups had just become available, and I became aware of the powerful methods of that theory. In particular, the most natural context for my problem on Cauchy sequences is that of a group acting on a totally ordered set. Such a group is not itself necessarily totally ordered (two-sided), but is (two-sided) lattice ordered by the naturally inherited pointwise order. Indeed, G. Birkhoff had observed, in his book on lattice theory, that the group of automorphisms of a totally ordered set is a natural example of a (usually, non-Abelian) lattice-ordered group, and F. Sik had investigated the structure of such automorphism groups. Using ideas based on my Cauchy sequence problem, I was able to show that every lattice ordered group is embeddable in the lattice-ordered group of automorphisms of a totally ordered set. I then found the obvious analogues in the context of lattice ordered groups for many of the basic results about general permutation groups. VII VUl Since then, these methods have become crucial in the study of non-Abelian lattice ordered groups. From the other direction, it was already pointed out in Wielandt's lecture notes that certain permutation groups (the "primitive but not strongly prim itive" ones) can be viewed as automorphism groups of a certain partial order on the permuted set. The pursuit of similar ideas has led a portion of the theory of permutation groups in the direction of viewing them as automorphism groups of certain structures on the permuted set. In the summer of 1993, a conference at Luminy in France, organized by M. Gi raudet, F. Lucas, and D. Gluschankof, provided the opportunity for many experts in ordered groups and in infinite permutation groups to explore the areas of overlap between the two subjects. The present volume is not, in the usual sense, a proceed ings of that conference; not all of the authors represented here were participants at the conference, and not all participants are represented. Rather, I have tried to choose a sample of articles by various authorities to represent a spectrum of ideas ranging from almost purely ordered groups at one end to almost purely permutation groups at the other. I hope that in this way the reader can get a good feeling for the interaction of the two subjects. The chapter in the present volume by V. M. Kopytov and N. Ya. Medvedev on quasi varieties of lattice-ordered groups, while mainly of a purely algebraic na ture, still contains many results whose proofs rely on the representation as ordered permutation groups. The structure theory of lattice-ordered permutation groups, much of it due to S. H. McCleary, is surveyed in McCleary's chapter in the present volume. In the chapter in this volume on Jordan groups by S. A. Adeleke, the permutation groups become automorphism groups of various order-like structures. From another standpoint, if G is the group of all automorphisms of some struc ture, then certain properties of the structure are reflected in G itself. In the best cases, one can "reconstruct" the structure from G. This is the subject of the chap ters in this volume by J. K. Truss (the structure is a totally ordered set and the group of interest is a certain quotient of the full automorphism group), M. Droste and R. Gobel (the structure is a McLain group), and especially the very general ap proach in the chapter of M. Rubin; to some extent, the same ideas are present also in the chapter by J. L. Alperin, J. Covington, and D. Macpherson (the structure is a quotient of an infinite symmetric group). The chapters of C. Praeger and of D. Macpherson are essentially of a permutation group-theoretic nature, but still use methods that are strongly reminiscent of those used in the more order-theoretic parts of the subject. To all of these authors, to the three organizers of the Luminy conference, and to several anonymous referees, I express my sincerest gratitude. W. Charles Holland In the Black Swamp August, 1995 Quasivarieties and Varieties of Lattice-Ordered Groups * V. M. Kopytov N. Va. Medvedev Mathematical Institute Altai State University Siberian Academy of Science Barnaul 656099 Novosibirsk 630090 Russia Russia 1 Introduction Many properties and statements of the theory of lattice-ordered groups (i-groups) can be formulated and proved in terms of first order logic. Special mention should be made of properties expressed by universal sentences such as identities and im plications, which can be referred to as the theory of varieties and quasivarieties, respectively, of i-groups. The theory of varieties of i-groups has been developed for more than two decades and contributions to it have been included in books and survey articles (Anderson and Feil [1], Kopytov and Medvedev [44, 45], Reily [63]). It was not until the mid-80s that the systematic investigation of the theory of qua sivarieties of i-groups began and the results obtained in this area are not available for wide audience yet. It is clear that the theory of quasi varieties is more general than that of varieties. Nevertheless, there have been obtained a number of results on quasivarieties of i-groups asserting that helpful and non-trivial properties of i groups can be defined by means of implications, and the theory of quasivarieties of i-groups itself is exciting and profound. This paper consists of two parts which are different in style and goal. The first part (sections 2 through 6) contains an introduction to the theory of quasivarieties of i-groups and presents the statements and proofs of those results whieh are, in the authors' opinion, the most interesting. The second part (sections 7 through 11) is a survey of results in the theory of i-varieties obtained after the publication of N. Reilly's paper [63](1989). The account in the present paper is "semi closed" since we assume the standard definitions and results on i-groups from the well-known ·This work was done with financial support of the Russian Fund of Fundamental Research (project code 93-011-1524) W. C. Holland (ed.), Ordered Groups and Infinite Permutation Groups, 1-28. © 1996 Kluwer Academic Publishers. 2 Kopytov and Medvedev books and survey articles, while the notions not previously defined are explained in the article. We employ standard notation and terminology of group theory (Hall [28], Kargapolov and Merzljakov [38]) and £-group theory (Reily [63]). PART 1: €-Quasivarieties 2 Generalities Let us recall that a formula <p of t.he first-order language and of the signature (J' is an implication (or quasi-identity) if <p is of the form (VXl, ... , Xn)((Al = Bl&A2 = B2&" . &Ak = Bk) => (A = B)) where AI,,'" Ak, Bl , ... , Ih, A, B are terms of signature (J' in free variables Xl, ... , Xn. The class K. of algebraic systems of signature (J' is a quasivariety if there is a set. cl> of implications such that an algebraic system G belongs to K. iff all implications of cl> are valid in G. An implication of signature £ = {. , -1, e , V, I\} is a formula <p of the predicate calculus of the form (VX!) ... (VXn)((Wl(Xl, ... , xn) = e& ... &Wk(Xl, ... , xn) = e) => => Wk+l(Xl, ... , xn) = e), where Wl(Xl, ... , xn), ... , Wk+l(Xl, ... , xn) are i-group words. An i-group G satis fies the implication <p if whenever then Wk+l(gl, ... ,gn) = e. A quasivariety of i-groups (or, simply, £-quasivariety) is the class X of all £ groups which satisfy a given set cl> of implications of the signature £ = {-, -1 , e, V, I\}. The set cl> is called a basis of the implications of the £-quasivariety X. From standard results in universal algebra (see Burris and Sankappanavar [7], Theorem 2.25.) we have the following characterization of i-quasi varieties: Theorem 2.1 Let .1:' be a class of i-groups. Then the following statements are equivalent: (1) .1:' is a quasivariety of £-groups, (2) X 1S dosed under i-isomorphisms, i-subgroups and reduced products, and contains a trivial i-group, (3) X is closed under £-isomorph1sms, i-subgroups, cartesian products and uitraproducts, and contazns a trivwl i-group.
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