Table Of ContentOrdered 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
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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.
