groups of auto morph isms of algebraic systems

8/3/2019 Groups of Auto Morph Isms of Algebraic Systems

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B. I.PlotkinProfessor of Mathematics, University of Riga

G roups of autom orphism s of algebraicsystems

translated byK. A. HirschProfessor of Mathematics, Queen Mary CollegeUniversity of London

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1o 0 3 6 9 6 2

WOLTERS-NOORDHOFF PUBLISHING 1972 GRONINGEN


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TABLE OF CONTENTS 1972Wolters-Noordhoff Publishing, Groningen, The Netherlands Preface.All rights reserved. No part of this publication may be reproduced, stored inretrieval system, or transmitted, in any form or by any means, electronicmechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

I General ProblemsChapter 1

Algebraic systemsISBN: 9001 71700 4Library of Congress Catalog Card Number: 72-151040 1.1 Algebras

Original edition published in 1966under the title "Gruppi avtomorfismovalgebraicheskikh sistem" at Moscow.

1.1.1 Definitions,1.1.2 Congruences and homomorphisms, 51.1.3 Direct and subdirect sums ofalgebras, 91.1.4 Freeand reduced freealgebras, 101.1.5 Commutative algebras. The commutator algebra, 131.1.6 Quasi-endomorphisms ofalgebras. Q-semigroups, 14

1.2 Multioperator groups1.2.1 Defini tions, 181.2.2 Near-rings. Multioperator near-rings, 201.2.3 Normal seriesand systems, 231.2.4 Abelianncss, nilpotency, solubility, 241.2.5 Direct decompositions. Complete reducibility, 261.2.6 Radicals inmultioperator groups, 31

1.3 Models. General algebraic systems1.3.1 Definitions, 361.3.2 The language ofthe lower predicate cal


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2.1.3 Transitive pairs, 672.1.4 Direct products ofpairs and other constructions, 722.1.5 Radicals connected with representations, 77

3.2 Decomposability, complete reducibility, imprimitivity 142

2.2 Existence theorems for representations 79

3.2.1 Preliminary remarks. Maschke's theorem, 1423.2.2 Clifford's theorem, 1473.2.3 Imprirnitivity and primitivity, 1493.2.4 The group of all automorphisms ofa completely reducible Q-group, 153

2.2.1 Preliminary remarks, 792.2.2 The theorem ofMostowski-Ehrenfeucht, 812.2.3 Atheorem ofBirkhoff, 862.2.4 Local theorems on representations, 872.2.5 Miscellany, 91

3.3 Stability and associated radicals 153

2.3.1 Galoisconnections for pairs, 932.3.2 Extension ofrepresentations. The Bourbaki scale, 952.3.3 More onGaloisconnections, 98

3.3.13.3.23.3.3

93 3.3.43.3.53.3.63.3.73.3.8

Finite stability. The y-radical, 153A remark onlocal boundedness ofa representation, 157Quasistabili ty. The fJ-radical, 159Onnilsetsin the acting group, 161Relations between thethree radicals, 164Radicals ofthedomain ofaction, 167General triangularity, 168Invertibility ofan endomorphism ofan Q-group, 170

2.3 Galois connections

2.4 Some outer properties and radicals . 101 3.4 Additional remarks on generalized modules 1732.4.1 Amethod of forming outer properties, 1012.4.2 Radicals of the acting group connected with distinguished properties, 1042.4.3 Almost periodicity and relative finiteness, 1062.4.4 Relative nilpotency and relative solubility, 1082.4.5 An example, 1112.4.6 Boundedness and algebraicity, 112

3.4.1 Annihilator properties, 1733.4.2 Annihilator properties and nil-properties, 1773.4.3 The Levitzkiradical ofan Q-near-ring, 180

2.5 Representations of groups and of Q-semigroups . 11 4

3.5 Automorphism groups of direct sums of Q-groups3.5.1 General remarks, 1823.5.2 Triangular groups of automorphisrns connected with direct decomposi-

tions, 183

182

2.5.1 Representations of Q-semigroups and generalized modules 1142.5.2 Q-semigroups associated with semigroups (or groups), 1172.5.3 Multioperator near-rings and representations on Q-groups, 1202.5.4 More on boundedness and algebraicity, 123

3.5.3 Generalized matrices, 186

Chapter 4Groups of automorphisms of vector spacesChapter 3

Automorphisms groups of multioperator groups3.1 Reducibility and irreducibility

3.1.1 Reducibility, 1273.1.2 Irreducibility, 1333.1.3 The radical ofarepresentation, 1353.1.4 The Jacobson radical ofan !2-near-ring, 1403.1.5 Generalizations, 141

127

4.1 Linear representations and linear pairs4.1.1 Introduction, 1914.1.2 Matrices and matrix representations, 1954.1.3 Triangularity and stability inlinear pairs, 2024.1.4 Finiteness conditions, 206

191

4.2 Finite-dimensional linear representations 2094.2.1 Introductory remarks, 209

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4.2.2 The local theorem, 2114.2.3 Finitely generated groups, 2134.2.4 Characters, 219

II Automorphism groups of groups4.3 Some properties of infin ite-dimensional l inear groups 226

Chapter 5

Some information from the abstract theory of groups4.3.1 Algebraic elements and local finite-dimensionality, 2264.3.2 Radicals ofa linear group and radicals ofits hull, 230 5.1 Group pairs 265

4.4 Groups of automorphisms of l inear algebraic systems 2355.1.1 Group pairs and semidirectproducts, 2655.1.2 The holomorph, 2675.1.3 Connections with the group ofinner automorphisms, 2705.1.4 Two remarks on reducibility ingroup pairs, 2724.4.1 Linear algebraic systems, 235

4.4.2 Algebraic linear groups, 2364.4.3 Regular automorphisms oflinear systems, 2434.4.4 Miscellany, 248

5.2 Radicals in groups 273

4,5.1 The fullsymmetric group, 2524,5,2 The fullmonomial group, 2594.5.3 The full linear group, A survey, 260

5.2.15,2.2

252 5.2.35.2.45.2.55.2.6

Radicals and ascendancy, 273Somespecificradicals, 277The Baerradical and the Gruenberg radical, 281Minimal radical classes, 286Locally bounded groups, 287The a-radical ef a group, Radicals connected with aninner pair, 289

4,5 The full linear group and the full symmetric group

5.3 Radical groups and W-groups 2905.3.1 Semisimplicity, The completely reducible radical, 2905.3.2 Plotkin groups and W-groups, 2925.3.3 A theorem onisomorphisms ofseries, 2955.3.4 Normal subgroups bounding the group, 298

5.4 Left Engel elements in groups. Examples 3025.4.1 The Hirsch-Plotkin radical and leftEngel elements, 3025.4.2 Left Engel groups. Right Engel elements, 3105.4.3 Examples, 312

Chapter 6Automorphisms of groups

6,1 Types of automorphisms 3246.1.1 Locally inner automorphisms, 3246.1.2 Outer automorphisms, 3256.1.3 Normal and central endomorph ismsand automorphisms, 325

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6.1.4 Nilautomorphisms, 3286.1.5 Regular automorphisms , 3286.1.6 Miscellany, 332

7.3.4 A ref inement for torsion-free groups , 3817.3.5 Examples, 383

6.2 Complete groups. The automorphism group tower 334 7.4 Sufficientconditions for stability 3866.2.1 Complete groups, 3346.2.2 Definition of the tower, 3366.2.3 Wielandt's theorem, 338

7.4.1 The der ived group and s tabil ity, 3867.4.2 The derived group and quasi stabi li ty , 3887.4.3 The role of the centre, 390

6.3 The automorphism group of the holomorph 343 7.5 Radicals of stable type in group pairs 3926.3.1 Completeness of the holomorph, 3436.3.2 The automorphism group of the holomorph, 345

7.5.1 Radicals of the acting group, 3927.5.2 Connect ions wi th the a-rad ical, 3987.5.3 Radicals in the domain of action, 400

6.4 Further questions . 3496.4 .1 More onquasi- rings generated byendomorphisms ofa group, 3496.4 .2 Residual f in it eness of automorphism groups , 350

Chapter 8

Finiteness conditions in group pairsChapter 7

Stability and nilpotency7.1 Finite stability and nilpotency 353

8.1 Stability and finitenessconditions8.1.1 The rank of a group, 4038.1.2 Finiteness of the rank of the domain of action, 4058.1.3 Finiteness of the rank or the acting group, 4098.1.4 Chain conditions in the acting group, 4118.1.5 Chain condit ions in the domain of act ion, 4148.1.6 More on relations between the (1- and p-radicals, 415

403

7.1.1 Introduction, 3537.1.2 The theorems of Kaluzhnin and Hall , 3547.1.3 Auxiliary lemmas, 3577.1.4 Addi tional remarks , 360 8.2 The operator case . 417

7.2.1 Statement of the problem, 3627.2 .2 Stabil ity and radical s of the domain ofaction , 3647.2.3 Relations between outer local n ilpotency and quasi stabi li ty of the acting

group, 3687.2.4 Examples, 370

8.2.1 General remarks, 4178.2.2 Another generalization of Zassenhaus' theorem, 4208.2.3 More on triangularity, 425

7.2 Quasistable and outer locallynilpotent groups of automorphisms 362

Chapter 9Automorphism groups of soluble and nilpotent groups

7.3 Someproperties of quasistable groups 3727.3.1 Pure automorphisrns, 3727.3.2 Almost periodic automorphisms , 3767.3.3 The existence of a central system ina quasi stable group of

automorphisms , 379

9.1 Automorphisms of Abelian groups 4289.1.1 Introduction, 4289.1.2 Periodic groups, 4289.1 .3 Torsion-free groups . References , 430

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9.2 Periodic, soluble and nilpotent linear groups 4 3 1 Preface.2.1 Periodic matrix groups, 4319.2.2 Solublegroups, 4379.2.3 Nilpotent matrix groups, 4419.2.4 Generalizations to theinfinite-dimensional case, 4469.2.5 Existence offaithful representations, 447

9 . 3 Groups of automorphisms of soluble groups 4 4 8

9.4 Automorphisms of nilpotent groups 4 5 8

Byan algebraic system, or a structure, wemean in this book, as is customarynowadays, a set together with a collection of algebraic operations definedon it.In recent years interest in various new types of algebraic systems has grown

very rapidly, and groups and rings are now only particular representativesof the whole family of such systems. Nevertheless, groups continue to occupya special position. The reasons for this are clear. Firstly, groups came intomathematics earlier than other abstract algebraic systems and have by nowreached the greatest degree of maturity. Secondly, a group structure is one ofthe constituent parts of many algebraic systems comprising, for example, thevery general notion of a multioperator group. And finally, what for us is ofthe utmost importance, the set of automorphisms of every individual algebra-ic, or more generally, mathematical structure is a group.The role of automorphisms iswellknown. In the study of all special mathe-

matical structures of an algebraic, geometric, or even metamathematicalorigin we come across various important kinds of symmetry. All these sym-metries can be analysed and described in the language of automorphisms.Therefore, the group of automorphisms of a specificmathematical structuredetermines in many respects the situation inside the structure itself . It iscommon knowledge that ever since the days of the Erlanger Programmautomorphism groups have been used to classify and systematize variousmathematical objects.Almost throughout the last century groups were interpreted in the first

instance as groups of automorphisms, and not until recently was it realizedthat a group can also be regarded abstractly, independently of the objects onwhich it acts. In every concrete group of automorphisms we have to distin-guish between internal, abstract properties and external properties connectedwith the object and the action. The abstract properties are important in thatthey are invariant under various concrete forms of the group, under variousisomorphic representations of it. The systematic investigation of abstract

9.3.1 Initial remarks, 4489.3.2 Automorphism groups ofsoluble Ai-groups, 4499.3.3 Complete groups ofautomorphisms, 4549.3.4 More on finitenessconditions ingroups, 455

9.4.1 Existence ofouter automorphisms, 4589.4.2 Miscellany, 4599.4.3 Automorphisms offree nilpotent groups, 461

Appendix 4 6 4Bibliography 4 7 7Subject index 4 9 6

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Preface Prefaceproperties and abstract classes of groups began with the appearance of theconcept of an abstract group. Nowadays the abstract theory of groupsexhibits a very rich and interesting inner life: the last few decades have wit-nessed the accumulation of a great many ideas and results concerning thespecific nature of the intrinsic structure of groups. Several important resultshave been achieved recently after many years of research, among them solu-tions of problems that so far we could only dream about.However, the internal development led to a certain estrangement: one

hardly sees the automorphism group in the group, and many ties are torn.On the other hand, in the investigation of specif ic transformation groups it isnot sufficient to take account of the achievements of the contemporary abstracttheory of groups. At present the position here changes drastically, andnumerous new links between abstract directions and the theory of transfor-mation groups have been forged. This interaction of the theory of groupswith the external world is of immense interest not only in branches of mathe-matics that make use of groups as automorphism groups, but also for grouptheory istelf. Suffice it to recall, for example, that many subtle facts of theasbtract theory of groups were fi rst obtained by external means, and the roleof these external methods grows rapidly.Here we may say that all these connections and interactions were guiding

principles of the original idea of our book. In fact, connections and inter-act ions are know to be one of the most remarkable aspects of every science.But it is perfectly obvious that a single book cannot possibly exhaust thistheme, and we had to restrict ourselves to a definite circle of questions. Wenow give a rough outline of the kind of problems with which the book isconcerned.First of all we remark that our orginal aim was a discussion of the auto-

morphism groups of groups and, to a lesser degree, of vector spaces - lineargroups, with attention focused mainly on abstract properties and applica-tions to abstract groups. Our starting point were to be the importantinvestigation of Mal'tsev, Baer and P. Hall, also Kaluzhnins by now wellknown and very simple theorem, which contains an external criterion for thenilpotency of automorphism groups; futhermore, research connected withthe study of abstract properties of linear groups and also papers concerningabstract properties of a group from the point of view of the existence ofsymmetries - automorphisms of one kind or another in it.On the other hand, bearing in mind that automorphism groups of groups

have much in common with linear groups and the theory of linear representa-

tions, we thought i t useful to consider these parallel facts from single pointof view. And so a new idea arose: to begin the book with general representa-tions and everything relevant to them. Here the word "general" has themeaning that we are concerned with representations of groups by automor-phisms of arbitrary algebraic systems, and not only of vector spaces, as thisis done in the classical theory of representations. But soon it became clearthat about half the book would have to be devoted to this beginning, and thisreally motivates the tit le of the present book. The author tried, in part icular ,to track down What parts of the classical representation theory carryoverto the more general case. Such investigations are interesting not only fortheir own sake: the common approach makes it possible sometimes to thrownew light on famil iar things, to unify various special constructions, and fre-quently to discover new links. Of course , working with general constructionswe must not lose sight of thei r provenance and thei r several concrete mani -festations.As a result, the book splits into two parts. In the first part various propo-

sit ions concerning the automorphism groups of several a lgebraic systems aregathered together. If we are given an algebraic system G, a group T and arepresentation of r by automorphisms of G (homomorphisms of r into thegroup of all automorphisms of G), then T becomes itself a group of auto-morphisms in which external properties manifest themselves. At the sametime we arrive at the concept of the pair (G, r) consist ing of G and r, where arepresentation of r by automorphisms of G is assumed to be fixed. Thisrepresentation determines an action of T in G, that is, a special mapping ofthe Cartesian product G x T into G, The general theory of automorphismgroups is, in essence, the theory of such pairs. The usual algebraic notions(subpairs , homomorphisms and isomorphisms, factor pairs , d irect productsetc.) can be defined, and the theory of pairs develops along the lines naturalfor algebra. In particular, thanks to the concepts of isomorphisms of pairswe can speak of abstract properties of pairs.Thus, we have three categories ofabstract properties: those of G and r, then

those of the pair: mutual properties of G and r, and those of the action itself.These three types of propert ies are closely interrelated . Already Galois theorydemonstrates how deep these links can be. Similar links can be traced inseveral other cases. Special attention is given in Part One to the case when thedomain of action G is a multi operator group. Here these multioperatorgroups, recently introduced by Higgins, turn out to be very convenientobjects for generalizations.

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Preface Preface

The whole Part One can be regarded as an introduction to the theory ofautomorphism groups of algebraic systems. An introduction - i f only for thefact that the most significant theories (but not the theorems!) assume a fairdegree of concretization. Of the deep theorems in Part One concerning thegeneral situation we mention, for example, the theorem of Mostowski andEhrenfeucht. Part One also has a chapter on the automorphism groups ofvector spaces: linear groups. On the one hand, this chapter supplements thepreceding ones, and on the other, it serves as an introduction to the generaltheory oflinear groups.Part Two deals with the automorphism groups of groups. The study of the

automorphism groups of groups is, of course, of great interest from thepoint of view of the abstract theory of groups. Here the corresponding pairsare called group pairs. A special role i s played by group pairs in which one ofthe groups, the left one, is a normal subgroup of the other and the representa-t ion is defined by transition to inner automorphisms. It is easy to see thatrelations between a group and its normal subgroups are convenientlyinvestigated in the language of such group pair-s, and the viewpoint of grouppairs often leads to a better understanding of a number of standard group-theoretical facts. Part Two also has a section on abstract properties oflineargroups.Both Part One and Two contain some remarks of a survey character that

are relevant to other specific cases. We do not speak in more detail abouttheir contents and only refer to the summary given above.From this summary it is evident that the book does not exhaust by any

means even the theme announced in its title. As a matter of fact, this themewould include Galois theory with all its generalizations and everythingconcerning permutation groups and linear groups. Galois theory is not in thebook al though th beginnings of the modern theory of automorphism groupsstem from this theory, from the discussion of automorphisms of fields.Linear groups and permutation groups only accupy a small part. The needfor a separate comprehensive book on the theory of linear groups andpermutation groups has been recognized long ago.We mention that the connections with topology are not treated inthe book.

Although these connections are of special importance for automorphismgroups, and although the introduction of a topology often proves useful evenin purely algebraic questions, the relevant topological facts reduce to somej nsignificant remarks.Furthermore, the book is not sufficiently complete as far as automorphism

groups of groups are concerned. Unfortunately we have to sta te that the bookcontains almost no material on the automorphism groups of specific systems.The results here are often very elegant, but such an account isolated from adetailed study of the corresponding objects is impossible.For similar reasons lit tle attention ispaid to the investigation ofthe position

of automorphism groups inside algebraic systems. We must emphasize distinct-ly that in this book the centre of gravity l ies on the side of the automorphismgroups and our main aim is to analyse how an abstract group functions as anautomorphism group and related questions which unify the general ideas wehave mentioned and, probably, the taste of the author. The Bibliography doesnot make up signi ficantly for thi s defect. In passing we note that it would bevery useful to have a book of more elementary character devoted to someconcrete external l inks of the theory of groups and written, say, in the spir it ofMal 'tsevs remarkable work [14].The majority of the results in our book are so far accessible in journal

papers only and no coherent account of them exists. Basically these are resultsof comparatively recent date, which have been obtained by many authors,among them the author of the present book. We have also included someunpublished results.The book is intended for readers interested in modern algebra and its

applications who are familiar with the elements of group theory. In a numberof cases we make use of some standard information on nilpotent and solublegroups. The entire group-theoretical material is, of course, in Kurosh's'Theory of Groups', and it is very desirable to be acquainted with this book,especially for Part Two.Part One is more independent. At the beginning of the book, in the first

introductory chapter, all the necessary information on general algebraicsystems is compiled, in part icular, on universal algebras, multioperatorsgroups, and models. The rudiments of the theory of universal algebras and ofgroups with multi operators are contained in another book by Kurosh'Lectures on general algebra',' However, having completeness of the expo-sition in mind. we recall here some facts that are available elsewhere. Allthese repeti tions only occupy a small part of the first chapter .We emphasize further that the notion of a model is used in the book not

1) Recently other monographs have appeared. We mention, for example, P. M. Cohn'sexcellentbook 'Universalalgebra',

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Preface PART ONEonly for the purpose of generalizat ions - the local theorem of model theory isused in an essential manner in a number of cases. This important theorem(due to Godel and Mal'tsev) has now numerous applications in algebra,particularl y in the theory of automorphism groups.Some open problems are stated in several parts of the book. Among them

there are quite specific unsolved questions, occasionally very attractive ones,it seems to us. We also mention problems connected with the development ofnew directions of research. Some of these are new.The Bibliography contains not only papers referred to, but also other

sources having a bearing on our theme. In this Bibl iography the order ofthereferences is the usual one. In references to propositions, sections and sub-sect ions the fi rst digit indicates the chapter, the second the section, the thirdthe subsection, and the fourth the number of the proposi tion inside the givensubsection.I have reported on individual chapters of the book in the Algebra Seminars

at Sverdlovsk, Moscow, Novosibirsk , and Riga.I am particularly grateful to the late academician A. I.Maltsev who hashad a deep influence on my work and has made a number of valuable

suggestions for this book. For many a useful advice I thank my teachersProfessors P. G. Kantorovich and A. G. Kurosh. I consider it my pleasantduty to express my thanks to V. G. Zhitomirskii, L. A. Simonyan and A. I.Tokarenko for their great help in the preparation of the manuscript forprinting. The editor of the book O. V. Dorofeev has made many usefulcomments.

General problems

The Author.

Preface to the English editionSome time has lapsed since the book was completed, and naturally newpapers on our topic have appeared. Unfortunately it was not possible toconsider all these papers and we had to confine ourselves to those that havean immediate bearing on the text. Compared with the Russian edition onlythe most necessary (and minimal) changes and additions have been made.Professor K. A. Hirsch has done me a great honour with this translation,

and I express my sincere gratitude to him.B. I. Plotkin

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CHAPTER 1Algebraic s ys tems

1.1. Algebras1.1.1. DEFINITIONS. As we have mentioned in the Preface, the number ofspecific types of algebraic systems has multiplied in recent years. Some ofthese have arisen in algebra itself, some in other branches of mathematics,in particular, in connection with the needs of geometry and mathematicallogic; not infrequently new algebraic systems come from physics, cybernet-ics, or technology. The theories of these systems contain many parallelfacts and constructions, so that i t is convenient to consider general algebraicsystems (structures) comprising all these special kinds. Such general algebraicsystems were first studied in the thirties, but the interest in them has grownconsiderably during the last decade.We begin with the defin it ion a (universal) algebra .Let G be a set. Every function of n variables that is defined on, and has

its values in, G is called an n-ary algebraic operation defined on G. If wis n-ary operation on a set G and aI' ab ... , an is an (ordered) family ofelements of this set, then the result of applying the operation w to thissystem of elements is denoted by a1 a2 allw.A set G together with a collection Q of algebraic operations defined on itis called an algebra (or a universal algebra). Sometimes we speak of anQ-algebra and in individual cases we use the notation < G, Q >, especiallywhen the system of operations Q has to be emphasized. The set G itself iscalled the basic set of the algebra, and the operations in Q the basic oper-ations.Among these basic operations of an algebra there may also be nullary

operations: constants. To use nullary operations is convenient in cases whenwe have to single out particular elements in an algebra. For example, agroup is an algebra with three basic operations: the binary operation ofmultiplication, the unary operation of t ransition to the inverse element , andthe nullary operation which singles out the unit element of the group. Al-though the various specific types of algebras are not tied by their origin tothe general concept of an algebra, the wish to see clearly the place of these

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1 Algebraic systems 1. I Algebrasconcrete systems in the combined realm of universal algebras is quite natural.There exist several devices of selecting concrete types or classes of algebras.They all hinge on the fact that the basic operations are subject to additionalrestrict ions of one kind or another. Often such restrictions have the charac-ter of ident ical relations. For example, groups, rings, and also certain otherimportant classes of algebras, can be specified in the language of identicalrelations.If G is an Q-algebra and H a non-empty subset of G that is closed under

all the operations in Q, then If relat ive to the system of operations Q formsa subalgebra of G. Every non-empty intersection of any set of subalgebrasis itself a subalgebra, so that we can speak of the subalgebra generated byan arbitrary subset of an algebra. If X is a subset of an algebra G, then wedenote the subalgebra it generates by {X } or {X}n . The subset X is calleda system of generators of the subaJgebra {X}.Next, we denote by Q the set of all 'complex' operations generated by the

operations in Q. The following proposition is obvious. IfH is the subalgebraof G generated by X , then the elements of H are all the elements of X andthe elements of the form: Xl Xz ... XnW' where XloX2, ... .x; EX and OJ i s anoperation in Q of the corresponding 'arity'.The compositum of a system of subalgebras of an algebra is defined in

the usual way. Clearly the system of all subalgebras of a given algebra(supplemented by the empty 'subalgebra', if some of the subalgebras havean empty intersection) forms a complete latt ice under the operation of inter-section and taking composita. We denote this lattice by ~(G).The notion of a local system of subalgebras is defined just as in the theory

of groups. A system [G a J of subalgebras of an algebra G is called a localsystem if a) the set-theoretical union of all the Ga is G, and b) any two mem-bers are contained in a third member of the system. It is immediately clearthat if an algebra G is generated by a subset X, then the system of all sub-algebras generated by fini te subset s of X is a local system in G, so that, inparticular we can find for every element g EGa finite subset X' cX suchthat g E {X '} . However, this also follows immediately from the remark madeabove on the representability of the elements of an algebra in terms of gener-ating elements.From the definition of a local system of subalgebras it also follows im-

mediately that every finite set of elements of an algebra belongs to somemember of a local system. This property is often taken as the definition ofa local system. In this context we observe that although in what follows we

start out from the definition given above, many results connected with localsystems turn out to be valid even under this weaker assumption.An algebra G is said to be Noetherian i f the lat tice of all i ts subalgebras

satisfies the maximal condition. This is easily seen to be the equivalent tothe postulate that the algebra itse lf and all i ts subalgebras have fin ite systemsof generators. An algebra having a local system of Noetherian subalgebrasis called locally Noetherian. In what follows we shall often have to assumethat an algebra under discussion is locally Noetherian. The simplest andmost important example of a locally Noetherian algebra is a vector spaceover a (not necessarily commutative) f ield. 'We now define the concepts of homomorphism, isomorphism, endomor-

phism, and automorphism of an algebra.Let G and G' be two algebras of equal type, which means that we may

assume the collection of basic operations Q to be one and the same for the twoalgebras. Next, let ffJbe a single-valued mapping of G onto G', where wewrite gffJ(or sometimes g " ' ) for the image of an element g E Gunder ffJWe say that ffJpreserves the n-ary operation OJ if the following perrnutabilitycondition holds: (ata2 ... anO)) ffJ= (at f f J ) ( a2f f J ) . . ( a n f f J ) OJ for arbitraryelements a, EG . The fact that ffJ commutes with the nullary operationsmeans that all the distinguished elements in G are carried into the corre-sponding distinguished elements in G'. The mapping ffJis called a homomor-phism of G if it preserves all the basic operations of the algebra. If, inaddition, ffJis bi-unique, then it is not hard to see that the inverse mappingffJ-1is also a homomorphism, and in this case q ) is caned an isomorphismof G and G'. Two algebras G and G' are called isomorphic i f there existsan isomorphism between them. We note here that we can also speak of ahomomorphic mapping of one algebra into another, that is, onto a sub-algebra of the latter. In cases where we have to emphasize that we areconcerned with a homomorphism onto the whole algebra we use the termepimorphism.The concept of isomorphism allows us to speak of abstract properties of

algebras: a property of an algebra is said to be abstract if the fact that analgebra G has this property implies that every algebra i somorphic to G also

1) In th is algebra the elements of the ground field are regarded as operations in their ownright , as operators. Of course, such an approach cannot do jus ti ce to the whole content ofthe theory of vector spaces. Vector spaces generally should be regarded as doubly-basedalgebraic systems (see 1.3.5).

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1 Algebraic systems 1.1 Algebrashas it. Correspondingly, a class S~of algebras is called abstract if S t containstogether with an algebra G also all the algebras isomorphic to G.We recall next that a transformation of a set G is a mapping of the set

into itself, and a permutation a one-to-one mapping onto itself. If a trans-formation of the basic set of an algebra G is a homomorphism, it is calledan endomorphism of the algebra. An automorphism is a permutation ofthe basic set that is also an isomorphism.The collection of all transformations af on arbitrary set G can be made

into a semigroup in a natural way (multiplication is the result of performingthe transformations in succession). Under this multiplication the coUectionof all permutations ofa set G form a group, which is called thefull symmet-ric group of G and is denoted by Sc.It is immediately clear that if G is an algebra, then the set of all its endo-

morphisms is a subsemigroup of the semigroup of all transformations of G,and the set of all automorphisms a subgroup of the group of permutationsof G. We denote the semi g roup of all endomorphisms of an algebra G by(f(G), and the group of all automorphisms by Il I(G) or Aut(G).For example, if G is a vector space, regarded as a universal algebra in thesense explained above, then the endomorphisms of G are the linear opera-tors, and the automorphisms are the invertible linear operators. Examplesof automorphisms of a group, semigroup, or ring G are the inner auto-morphisms. Interesting examples of automorphisms occur in the theory offields. Concerning commutative and non-commutative fie lds we should notethat they are distinguished in the class of rings by the addition of one unaryoperation, which associates with every non-zero element its inverse. This isa partial operation, so that we can speak of fields as universal algebras onlywith the corresponding reservation.Various specific examples of automorphisms and of groups of automor-phisms can be found in the book by A. G. Kurosh [4J 'Lectures on General

Algebra' and in other algebraic textbooks. We recommend the reader toacquire a good stock of such examples, and among them also for systemswith relations (see 1.1.3), so as to have solid ground for general construe-tions.A subalgebra H is caned characteristic if ha E fl for arbitrary h e 11 and

(J E I}l(G). If hip E H for every h e H and every q ; E G :( G), then H is calledfully invariant. All characteristic subalgebras of a given algebra G form asublattice in the lattice of all subalgebras of G, which we denote by Z(G).The fully invariant subalgebras of G also form a sublattice. The lattice Z(G)

characterizes in a certain sense the degree of homogeneity of G: an algebraG is said to be homogeneous if Z(G) consists only of G itself, or possibly,also of the subalgebra O( G) generated by all dist inguished (or null) elements,provided that they exist . Clearly O(G) is always a characteris tic subalgebra.Often homogeneity of an algebra is understood as the postulate that thegroup of all its automorphisms acts transitively on the set of all non-nullelements of the algebra. This kind of homogeneity is a fairly rare phenome-non (see, for example, the paper by Kostrikin [lJ the main result of whichwill also be stated in 4.4.4).A subalgebra H of an algebra G is called strictly characteristic if H is a

fixed element under the group of all automorphisms of the lattice Sf(G) ofall subalgebras of G. Every strictly characteritic subalgebra is characteristic,but the converse is not true, in general.We write 1 f { ( G ) for Aut(Sf(G)), the group of all automorphisms of the

lattice Sf(G). Every automorphism (J of an algebra G induces some elementof Ill(G), which we denote by ii . The mapping (J - - - > (j is a homomorphismwhose kernel consists of those automorphisms of G that leave every sub-algebra of G fixed.

1.1.2. CONGRUENCESANDHOMOMORPHISMS.In the homomorphism theo-rem to be proved below the essential role is played by the concept of a con-gruence of an algebra. An equivalence on a set G is, of course, a binaryrelation 011 G having the properties of reflexivity, symmetry and transitivity.Every equivalence p determines a partition of G into disjoint equivalenceclasses X a (two elements a and b of G are equivalent if and only if theybelong to one and the same equivalence class X o J The fact that a and bare equivalent is denote by apb. Every equivalence class X a is uniquelydetermined by any representative: if x is an element of X a' we write X a ==xJ (or [ x J p ) .Next , the classes of an equivalence p can be regarded as elements of a

new set, which is called the factor set of G with respect to p and is denotedby G/p.Let w be an n-ary operation on a set G. An equivalence p is called a con-

gruence relat ive to this operation if for arbitrary a.; ai E G, i=1, 2, , n,the fact that aipa; for every i implies that (a1aZ anw) p(a;a~ a~w).When Gis an Q-algebra, an equivalence p of the basic set is called a con-gruence of G if p is a congruence relative to each operation in Q.

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J Algebraic systems 1.1 AlgebrasTo every congruence of an algebra G there corresponds a natural homo-

morphism of the algebra. Let P be a congruence on G and Gl p the corre-sponding factor set. Let c o be an n-ary operation in Q and [alJ, [a2J, ... ,[anJ a family of n elements of Glo,Setting

If Pl and pz are two binary relations on a set G, their product P l P Z is thebinary relat ion defined by the rule: X P l P Z Y if and only if for some Z we havex p 1 z and Z p z y . The relations Pl and p z are said to commute if P I P 2 =PZPl.Now let P e t ' lJ. E 1, be a collection of binary relations. The compositum of

all these relat ions is the relat ion P defined by the following rule: X P Y meansthat we can find al, a z , . . . , all E 1 and Z 1 ' Zz, ... , Z , , _ I E G such that x P a l z 1 ,z 1 P a zz 2 , . . . , z n - 1 P a " Y It is easy to verify that the compositum of equivalences is also an equiv-alence, and the compositum of congruences in the new sense is a congruencewhich coincides with the earlier defined composi tum of congruences. If twocongruences PI and pz commute, their compositum is the same as theirproduct P 1 P 2 . Under the operations of intersection and compositum all thecongruences of an algebra G form a complete lattice which we denote by:.f'(G). In Birkhoff's book [3 J it is shown that if all the congruences of analgebra G commute in pairs, then :.f'(G) is modular lattice.

we define c o as an operation on GIp. The significance of the concept of acongruence consists obviously in the fact that the operation so defined doesnot depend on the choice of the representatives of the equivalence classes.It is now clear that all the operations of Q automatically carryover to theset of equivalence classes and that this set becomes an Q-algebra, which wecall the factor algebra of G with respect to p. By associating with everyx E G the equivalence class [xJ we obtain a natural homomorphism of Gonto the factor algebra Glp, which we denote by ( { J p .Now let u: G -> G' be a homomorphism of G onto another algebra G'.We write xpy, X ,Y E G,if xu ~= yu. It is easy to verify that P is a congruence

of G. We associate with every class [x J of this congruence the elementxu. EG', and denote this mapping by 1 j J . An immediate check shows that Ij Jis an isomorphism of the algebras G I p and G' and that ( { J p l j J =l. All wehave said forms the content of the homomorphism theorem for algebras.This theorem shows that the homomorphisms of an algebra stand in aone-to-one correspondence with their congruences.it is natural to introduce lattice operations for the congruences of an

algebra. We recall, first of all, the definition of the intersection of equiva-lences. Let P =X p J be a collection of equivalences of an algebra G. Theintersection of all the equivalences of this collection is the equivalence Pdefined as follows: if x E G, we denote by [x J P the intersection (over all a)of all X p containing x. This set of all [xl, determines, in fact, an equivalence.It is easy to see that the intersection of any collection of congruences of analgebra is also a congruence.Next, if P i X ' a E 1, is again a collection of congruences in G, then its

compositum is naturally defined as the intersection of all congruences of Gcontaining all the P a . Here a congruence P iX belongs to P (in symbols P C ! c )if X P a Y always implies that X P y. There is another more constructive approachto the definition of the compositum, which based on the operation of mult i-plication of binary relations.

Now let P be a congruence of an algebra G and fl a homomorphism of itonto an algebra GI'. If [xJ, x E G, are the congruence classes of p, then theirimages [XJI ' in GI' cover, but do not necessarily split , GI', that is, these [xyneed not form an equivalence in Gs. In the good case when all these [xyform an equivalence on GI', we say that the congruence P and the homomor-phism fl are compatible. If this compatibility holds, we denote the corre-sponding equivalence by p" , It is easy to see that here pi ' is automatical ly acongruence of GI'.Of course , for groups, and consequently for Q-groups (see 1.2), the special

stipulations on compatibility are superfluous, but already for semigroupsthey are necessary. Furthermore, it is not hard to check that when thecongruence of the homomorphism fl belongs to p, the required compatibilityholds.In the general case, when p and fl are not necessarily compatible, we

proceed as follows, If n is the congruence of the homomorphism ,u and{p, n} the composi tum of p and n, then clearly {p, n} and tl are compatible.Now we write pI' =p, nY . Let us show that if p and It are compatible,then this new p i" is the same as the one defined earlier. It is sufficient toestablish that in the presence of compatibility Xll{p, n}"i' implies thatxllpllyl'. Suppose that xl' {p, n}"yl l. Then x{p, n}y. The latter means thatthere exist Zt> Z2' . , Zn such that

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1 Algebraic systems 1.1 Algebrasof G into individual elements, and the unit congruence, which consists of asingle class, namely G i tself. An algebra G is called simple if it has no non-trivial congruences, and characteristically simple if it has no non-trivialcharacteristic congruences. It is easy to see that for groups characteristicsimplici ty and homogeneity are equivalent concepts.As we know, every automorphism (J of an algebra G carries every con-gruence p of the algebra into some other congruence p" : Consequently, (J

induces a permutation (j on 2"( G ). Iti s easy to see that, in fact, the mapping(J --* (j is a homomorphism of I}{(G) into the group of all automorphisms of2"(G). The kernel of this homomorphism is the collection of all thoseautomorphisms of G that leave every congruence fixed. (In this context seealso the paper by Dwinger [lJ).

where each of the equivalences involved is either (J or tt: When we now goover the images of all these elements in GI', then at those places where poccurs we have to replace it by p" , and where n occurs, we obtain an equality.Hence we can go over everywhere to the equivalence p" and, consequentlyobtain Xl'pl'yll.Thus, if G is an algebra and p one of its congruences, then to every

epimorphism u: G --* GI ' there corresponds a congruence p" of G".Next , if A is a subalgebra of G, let A n p stand for the equivalence in A

whose classes are the intersections of the classes of p with A. It is easy tosee that A n p is a congruence in A, and that the factor algebra AI A n pis isomorphic to a subalgebra of Glo,We need the following definition. If /1 is an endomorphism of an algebra

G and pone of its congruences, we call his congruence invariant under /1if xp y always implies that x"p yll. This definition of invariance is equivalentto the following: every equivalence class of p" belongs to some equivalenceclass of p. For suppose that p in invariant under the endomorphism /1 , thattt is the congruence of /1 , and that X is a congruence class of {p , n}. Wehave to show that if x and yare two elements of X, then xllpyl'. The factx and y belong to the same class X means that there exist Z1> Z2' .. , Znfor which

1.1.3. DIRECT AND SUBDIRECT SUMS OF ALGEBRAS. Let [Gu.] be a familyof Q-algebras of equal type, the indices exbelonging to a set 1. We denoteby G the set whose elements are all the functions that associate with everyexE 1 a well-defined element a ; in Ga . We write a =au.) for such a function.The element aU . is called the component of a in Ga..The operations of Q extend to G in the following way. Let w be an n-ary

operation in Q and a(k) =a~), k =,2, ... , 11 , be n elements of G. Then weset:

(1) (2) (n)_ ( 1 2 n )a a a (J) -- aU .aa .. a"w .where, as above each of the equivalences is p or ti, When we now go overto the images under f1 . and use the invariance of p, we find: The so-defined Q algebra Gis called the complete direct sum of thealgebras Ga.

If all the Go: have one-element subalgebras, we can also speak of thediscrete direct sum of all the Ga . This is the subalgebra in the completedirect sum of the Ga' exE1, that consis ts of the sequences ( a u . ) in which onlyfinitely many components fail to belong to the corresponding one-elementsubalgebras.An important role is played by the concept of a subdi rect sum of algebras,

which goes back to Remak. A subalgebra (fj of the complete direct sum ofalgebras Ga is called a subdirect sum of these algebras i f for every exE 1 thecomponents of the elements in (fj with the index ex range over the wholealgebra Ga . If < f> is a subdirect sum of algebras Ga, then by associating withevery a E (fj its component a" we obtain a homomorphism of (I) onto Gx'

And so we have xl'py". The converse is obvious.It is not hard to see that when (J is an automorphism of an algebra G,

invariance of p under (J and (J-1comes to the same thing as that p and p"coincide.A congruence p is said to be characteristic if it is invariant under an

automorphisms of G, and [ully invariant if it is invariant under all endo-morphisms.Two congruences are called trivial: the null congruence - the splitting

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1 Algebraic systems 1.1 Algebras

G = G (Q , X ).

then for any n words Wi> W2, . , Wn the word WI W2 .. WnW is the resultof applying the operation W to these words. This algebra of words is saidto befree or, more accurately, absolutely free.It is easy to find all the automorphisms of the algebra G (Q , X ).Let (J be a permutation on X, W an element of G, and suppose that in

the expression of this element the following elements of X take part: Xl'Xz, ... , Xk When we replace in wall the these X, respectively, by the elementsXl (J, X2G', , Xk(J, we obtain a new word, which we denote by W(J. It iseasy to see that the mapping W - - - > W(J is an automorphism of G.On the other hand, suppose that qJ is an automorphism of G (Q , X ).

Bearing in mind the fact that the mapping qJ must preserve the st ructure ofwords, we conclude that X remains invariant under qJ . Therefore to theautomorphism qJ there corresponds a permutation of X. This correspondencebetween the elements of the group of all permutat ions of X and the elementsof the group of automorphisms of G(Q, X) is obviously an isomorphism.Proceeding now to the concept of a reduced free algebra, we begin by

making the definition of an identical relation more precise. Let X and Q beas above, and let W I and W2 be two words. Suppose that Xl' x2, , Xk areall the element of X that occur in the expression of the words WI and W2,and that 01, O 2, , 01 are the symbols of the nullary operations in Q inthese words. We say that the identical relation Wj =W2 is satisfied in anQ-algebra G if this equation holds in G when all the Xi' i= 1, 2, ... , k, arereplaced by arbitrary elements of G and the relevant distinguished elementsof the algebra are substituted for 01, Oz, ... , 01,Of course, identical relations can be regarded as independent objects,without connecting them with a defin ite Q algebra.We now choose a system A of identical relat ions and write S\ for the class

of all algebras in which all the relations of A hold. Every such class S\ ofalgebras is called primitive. An easy verif ication shows that every primitiveclass of algebras is closed under complete direct sums, homomorphisms,and the operation of taking subalgebras in an algebra. Due to these impor-tant properties primitive classes of algebras playa special role. On the otherhand, according to a remarkable theorem of Birkhoff [3 J (see also theAppendix) every class of algebras that is closed under the three operat ionslisted is primitive. Thus, primitive classes, which are also called variet ies,have a good invariant characterization.Next, we form for every system A of identical relations a certain con-

gruence of the algebra of words G (Q , X ). First of all, we denote by A the

To this homomorphism there corresponds a congruence Pa . It is easy to seethat the intersection of all these Pa ' c: E 1, is the null congruence in G. Thefollowing converse proposition also holds:1.1.3.1. If in an algebra (fj a system of congruences p ; with the inter-section P is given, then the factor algebra (fj/p is isomorphic to a subdirectsum of the factor algebras l/Pa'For let G be the complete direct sum of the f f i/ P I X ' With the element a E (fjwe associate the element ii E G so that its component Ci a in (fj/Pa is thecongruence class of Pa containing a. It is easy to see that this mapping is ahomomorphism of (\'j into G. The congruence of this homomorphism is theintersection of all the Pa ' that is, p. It is also obvious that the componentswith index o: of the elements of (fj/p range over the whole algebra ffi/Pa'In parallel to this theorem, a number of papers give conditions under

which a system of congruences Pa of an algebra G is an algebra isomorphicto the complete (or discrete) direct sum of the factor algebras G/ P I X "

also mention that just as, for example, in rings (see Jacobson [4]),there is a natural way of defining a topology in the complete direct sum,which plays an important role.

1.1.4. FREE AND REDUCED FREE ALGEBRAS. Let X be a set and Q acollection of algebraic operations. Strict ly speaking, we are here concernedwith symbols of operations, because operations are not yet defined. Thenotion of a word is defined inductively. Every element of X and everysymbol of a nullary operation is regarded as a word. Next,if the expressionsWI' Wz, .. , Wn are already defined as words and if ill is the symbol of ann-ary operation in Q, n ;? ; 1, then the formal expression W1W2 WnW isalso regarded as a word, So we have defined a collection of words, whichwe denote by

The set G can be treated in the following way as an algebra under the systemof operations Q. Every nullary operation in Q selects the symbol of thecorresponding nullary operation. If t is an n-ary operation in Q with n ;? ; 1,10 11


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1 Algebraic systems J. J Algebrasnew system of relations that is obtained from A in the following way: if therelation Wj =W2 belongs to A, then by replacing in it the elements of X byarbitrary words, we obtain a relation w~=w~, which belongs to X Wedefine an equivalence transformation of words as a transformation of thefollowing form: if v is a word and w a subword such that the relat ion W =w'holds in X, then by replacing in v the part w by w ' we obtain a new word o'Two words v! and V2 are said to be equivalent if we can go from one to theother by applying finitely many equivalence transformations.It is not hard to see that the so-defined equivalence of words is, in fact,

a congruence of the algebra G (Q , X ) and that this congruence is fully in-variant. The factor algebra of the algebra of words by such a congruenceis called a reduced free algebra or afree algebra of the primitive class S tdefined by the system of identies A. We denote this free algebra by G. It iseasy to see that G also belongs to S\ and, that furthermore, the followingcharacteristic property holds: every mapping II: X -> G' of X into an algebraG' belonging to 5t can be extended to a homomorphism It: G --> G'. Hencewe obtain immediately the following theorem:

Next we note that in the theory of universal algebras, just as in the theoryof groups, an important role is played by free products and reduced freeproducts (in one sense or another). Concerning the general theory of suchoperations see, in part icular , the paper by Baranovich [1J . which continuesthe group-theoretical investigations of Golovin [J J .Finally we mention that Valutse [lJ investigates the semigroup of allendomorphisms of reduced free algebras.

1.1.5. COMMUTATIVEALGEBRAS.THE COMMUTATORLGEBRA. Let oi, beann-ary and (1)2 an m-ary operations of a system Q, n ~ 1, In ~ 1, and let (xij)i =1, 2, ... , m, j=1, 2, ... , 11 , be a matrix formed from elements of aset X. We write:

1.1.4.1. Every algebra G of a primitive class 5~ is a homomorphic imageof a free algebra o j" this class.

We call w! =Wz the relat ion of permutabili ty of the operations Wl and CO2 .Two operations W1 and Wz are said to commute on an Q-algebra G if therelation of permutability for these operations holds identically on G. Wecan also speak of permutability of two operations of which one is nullary.This means that if 0 is a nullary and co an n-ary (n ~ 1) operation, thefollowing condition holds: 00 ... Ow = .

"----v-----'n times

Now let A be a system of identical relations defining a primitive class .H ,and let G be an algebra. We write peA, G) for the congruence G that coin-cides with the intersection of all congruences p for which G / p is an algebrain R We know that G/p(A, G) also belongs to R The congruence peA, G)is called the A-congruence in G or the congruence defined by te system A.

1.1.4.2. Every A-congruence of an algebra G isfully invariant.

Permutability of two nullary operations means that the correspondingdistinguished elements of the algebra are the same. An algebra G is calledcommutative if any two of its operations, distinct or equal, commute on it.Hence it follows, in particular, that in a commutative algebra we cannothave more than one distinguished element, which we call its zero. Clearlythe class of all commutative Q-algebras is primitive. It is easy to see that inthe case of groups this definition leads to Abelian groups.' Starting from the

This congruence generalizes the concept of a verbal subgroup inthe theory of groups. A simple check shows that if II is a homomorphismof G, then we have the formula: peA , G il) =eA , G )" .The following theorem holds:

Let peA, G) = be a A-congruence in G and (J an endomorphism of thisalgebra. Since all the relations of A hold in Ga/p Il Ga, the congruencepeA , G ") belongs to p Il G" . When we now take into account that peA , Ga ) ==lY , we obtained the required proposition.

1) An operation w is said to be symmetr ical on a given algebra if every permutation ofarguments leaves the corresponding value unchanged. For groups symmetry of multipli-cation and perrnutability with itself turn out to be equivalent conditions. Wecould indicatea number of other classes of algebras with a similar situation.

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1 Algebraic systems J. l Algebrasanalogy with groups it makes sense to introduce the following special typeof A-congruences.Let G be an algebra. We define the commutator algebra by the congruence

p e G ) that coincides with the intersection of all congruences of G whosefactor algebras are commutative. Obviously G/p(G) is a commutativealgebra.

plicative semigroup of K, which we denote by D(K). The following pro-posi tion is easy to prove:1.1.6.1. If Lis a sub-semigrou.p of D(K), then the subatgebra of K gener-ated by the elements of Li s closed under multiplication.

L1.6. QUASI-ENDOMORPHlSMS OF ALGEBRAS. Q-SEMIGROUPS. Let H(G) = Hbe the semigroup of all transformations of the basic set of an Q-algebraG. We show that all the operations of Q carryover naturally to H.For example, let W be an n-ary operation in Q and tp" tpz , .. . , tpn dementsof H. Then tpl t pz .. . tpnW is the transformation defined by the formula:g( t pl t pz . .. tpJlw) = = c tpl)(gtp2) .. . (gtpn)w for every g E G.

If 0" EQ is a nullary operation, then we assume in the same vein that isselects in H the transformation ell that carries every g EG into the elementof G selected byIn this way H becomes an Q-algebra and, in addition, we have a multi-plication in which is easily seen to have the property of left distributioit yover all the operations of Q: if co is an n-ary operation in Q and ~?b tpz,. .. , tpn EH, then

First of all we observe: an induction shows immediately that in an Q-semi-group left distributivity holds not only for the operations of fl, but also forall derived operations - the operations in Q. The same can be said of rightdistributivity for the elements of D(K). Now let U and v be two elements of{ L } Q and suppose that we have found for them

u v =~u ( v 1 V Z . . . v m W Z ) =u v j ) ( u v z ) . . . ( u v m )W Z ==( U 1 V 1) ( U Z V 1) ' . . (UIlVl)Wj)((U1Vl)(tl, V Z)(u 2V Z) . . . ( U n V 2)W ,) , . . ( ( u ,v " , ) ( U Z vm) . (U nV m) W l ) W 2 ,

so that u u is also an element of {qQ. The case when there are distinguishedelements among the factors is equally easy to deal with. This shows that theQ subalgebra generated by he elements of L is, in fact, even an Q-sub-semigroup.In particular, the subalgebra generated by all the endomorphisms of an

algebra G is an Q-subsemigroup of H. The elements of this subalgebra arecalled quasi-endomorphisms of G, and we denote here by H' (G ) the set ofall quasi-endomorphisms.

Right distributivity does not hold in general, but clearly holds for trans-formations rp that are endomorphisms of G. For a nullary operation 0" therole of lef t distributivity is played by the universally valid equation q J e a = e a,tp EH, and right distributiv ity is equivalent to the condition s,tp = ea 'We now make the following definition.A set K is called an Q-semigroup if all the operations of Q are defined

on it (K is an Q-algebra), and also a multiplication under which K is asemigroup (the multiplicative semigroup of K), where the multiplication isleft distributive over all operations of Q.

The next theorem is a generalization of a well-known fact concerningendomorph isms of abelian groups.An Q-semigroup is said to be d i s t r i b u t i v e if each of its elements is distrib-

utive.

The distributive elements of an Q-semigroup are those for which not onlythe left, but also the right distributive law is satisfied. The collection of alldistributive elements of an Q-semigroup K is a sub-semigroup of the multi-

1.1.6.2. If G is a commutative Q-algebra, then the set of all its endomor-phisms is a distributive Q-semigroup.We need only show that each element of H'(G) is an endomorphism of G.

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1 Algebraic systemsTo prove this fact it is enough to establish that the set of all endomorphismsis closed under the operations in Q.Let w be an n-ary operation in Q and r p l ' r p z , . . . , r p " endomorphisms

of G. We show that l f J l r p 2 . . . r p n W is also an endomorphism. Let to' be anm-ary operation in Q and a i' a2, , am elements of G. Using the fact thatwand co ' commute on G, we obtain:

a la 2 a m W '( r p l ~ J 2 ' " r p n w ) =c = (( a 1a 2 a m w ' ) r p l ( a t a Z amw ') r p 2 . . . . . ( a la 2 . . . a m w ' ) rp n) w = (( a l r p l ) ( a 2 r p l ) ' . . . . ( a m rp J) w ' ( a l r p Z ) ( a 2 r p 2 ) ' . . ( a m r p z ) w ' . . . .. (at r pn )( az r pn ) . . ( am rp n) w ' ) w = ((a l r p 1 ) ( a t rp2 )' .. . . (a l r pll )w (a 2r pl )( aZrp2)' .. (a 2r p l l ) w , . . ( amr p t ) ( amr p 2 ) ' . . . . ( am rp n) w ) w ' = (al( r pt r p2 . .. r p n w ( a z ( r pl r p2 . .. r p n w . . . . . ( a r n ( rpl r p 2 . . . r p n w w '.

Furthermore, if 0 is the zero element of G, then

AU this means that rp 1 r p2 . .. ' P n ( J ) is an endomorphism. It is also evidentthat the zero element of H'(G) is an endomorphism.Of course, in the general case H'(G) is not distributive, and not every

quasi-endomorphism is an endomorphism. In what follows we have tosingle out a class of Q semigroups which, like l-I'(G), have distributive Q_generators. The presence of such generators facilit ates the investigation ofQ-semigroups. By computations similar to those made just now we canprove the following proposition:1.1.6.3. If K is a commutative Q-algebra and K has as an Q-semigroupdistributive generators, then K is a distributive Q-semigroup.In this context see also the paper by Evans [1].It is easy to see that a homomorphic image of an Q-semigroup is also an

Q-semigroup, and that distributive elements go under a homomorphism intodistributive elements. We now prove the following proposit ion:1.1.6.4. If K is an Q-semigroup with distributive generators and p an16

1./ A lgebrascongruence of K as an Q-algebra, def ined by a system of identical relations,then p is also a congruence of K as an Q-semigroup.As we know from 1.1.4.2, p is a fully invariant congruence of K as anQ-algebra. Now if u is an arbitrary element of K, then by left distributivitythe mapping x -+ UX, X EK, is an endomorphism of the Q-algebra K.Hence, using the fact that p is fully invariant , we conclude that xp y implies(ux)p(uy). Similarly we obtain for a distr ibutive element u that xp y implies(xu) p (yu). Since K has distributive generators, the latter property holdsfor all u EK.Suppose now that xpy and upv. From the preceding remarks we have

(ux)p(uy) and (uy)p(vy). Hence (ux)p(vy), as required.Some problems in the theory of Q-semigroups are discussed in the paper

by Khion [1]. He investigates, in particular, the adjunction of a unit elementto an Q-semigroup with preservat ion of one condition or another. It turnsthat this matter is by no means simple.Next we examine the case when two Q-algebras A and B are given. Let

HCA, B) be the set of all single-valued mappings of A into B. Just as thiswas done for H ( G ), all the operations of Q can be carried over naturally toR (A , B ). For example, if (J ) is an n-ary operation in Q and r p l ' r p 2 ' . . . , r p "are elements of R(A, B) , then rp l r p 2 r p " w is the mapping defined by theformula: a ( ' P l r p 2 '" r p n W ) = ( a r p l ) ( a r p n )w for all aEA. So we obtainan Q-algebra RCA, B ). It follows immediately from the definition that forevery a E A the mapping rp -+ a rp is a homomorphism of H(A , B) into B.Let P u be the congruence of this homomorphism: r p p a V l means that a r p ~ ~ m p .Clearly the intersection of all Pm a E A, is the null congruence of theQ-algebra R (A , B ). Hence and from Theorem 1.1.3.1 on subdirect sums ofalgebras it follows that J- l (A, B) belongs to every primitive class of algebrascontaining B. A similar proposition is applicable, in particular, to theQ-algebra H ( G ).An immediate verification establishes the following property:

1.1.6.5. If B is a commutative algebra, then the set of all homomorphismof A into Bforms a subalgebra of H( A, B).This algebra is usually denoted by Hom(A, B).

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1 Algebraic systems 1.2 Multioperator groups1.2. Multioperator groups the set of operations Q and the operations of the additive group, and among

them the nullary operation that selects the zero element. This zero by itselfforms an Q-subgroup. Clearly by an Q-subgroup we have to understand asubset II c G that is a subgroup of G+ and a subalgebra of G, as an algebra.It is easy to see that a homomorphic image of an Q-group is also an Q-group.Next, let q ) be a homomorphic mapping of an Q-group G into an Q-groupG', and let p be the corresponding congruence. Let H stand for the completeinverse image in G of the zero of G'+. Since rp isa t the same time a homomor-phism of additive groups, H is a normal subgroup of , and the co sets ofH exhaust a ll the congruences classes of p.Moreover, H has the following property: for every n-ary operation w EQ

and arbitrary Xl> X2, , XIl E G, h1 ' hz, ... , 11"E 11 we have

1.2.1. DEFINITIONS. It is well known that in the case of groups,rings,' and in a number of other cases, homomorphisms have a remarkablefeature: they have kernels, which make it possible to realize the classes ofthe corresponding congruences as classes with respect to these kernels. Asimilar si tuation prevails for the types of algebras that are combined underthe concept of a group with a system of multioperators. Groups and ringsare special forms of groups with multioperators, The theory of such multi-operator groups was created by Higgins [1], and a fairly detailed accountis in the book by Kurosh [4J.We now give the fundamental concept of this theory.Let G be an algebra under a system Q of operators (multioperators) and

suppose that , furthermore, on G a group structure is defined relative to anaddition, which is, in general, non-comm utative (the additive group of G).This set G, together with the group operations and the system of operationsQ, is called a group with multioperators (or a multioperator group), pro-vided that for every O J EQ the following condit ion holds: 00 ... Ow = .Here 0 is the zero of the additive group of G ;:md is repeated n t imes if wis an n-ary operation.A multioperator group is frequently cal led an Q-group so as to emphasize

the system of operators Q. The additive group of an Q-group G is alsodenoted by G+.

(1 )

1) Rings are not, as a rule, assumed to be associative.

Every normal subgroup H of the additive group of G+ having this propertyis called an ideal of the Q-group. If in (1) we take all the elements Xi as zero,we find that an ideal is always an Q.-subgroup. (1) also shows that theclasses with respect to each ideal form a congruence of the Q-group so thatideals of an Q-group are kernels of homomorphisms. Owing to the fact thatcongruences of Q-groups are in one-to-one correspondence with ideals weare now in a position to replace the notation for the Q-factor group G Ip byGIH , where H is the ideal corresponding to the congruence p.Itis not hard to see that the intersection of any set of ideals of an Q-group

is again an ideal and that the following rules hold: if P a is a collection ofcongruences of a multi operator group and H a are the ideals correspondingto these congruences, then the ideal corresponding to the intersect ion of allp" i s the intersection of all the H a. The compositum of ideals is an ideal andcoincides with the subgroup of the additive group G+ that is generated bythe additive group of the original ideals.It is also easy to verify that the compositum of congruences of an Q-group

is associated with the compositum of the corresponding ideals. Hence itfol lows that the lattice of all congruences of an Q-group G is isomorphic tothe lattice of ideals and, consequently, that it can be regarded as a sub-lattice in the lattice of all Q-subgroups of G. The homomorphism (see 1.l.2)of 1 2 1 ( G ) into the group of all automorphisms of the lattice '(G) now turnsinto a homomorphism of \ 1 ( ( G) into the group of all automorphisms of the

It is easy to see how groups and rings fall under the general definition ofmulti operator groups. Vector spaces are also Q-groups: here the system Qconsists of the scalar operators of the ground field (commutative or not).It i s altogether clear that Q-groups are to a certain extent a generalizationof rings, except that addition is not necessarily commutative and that in-stead of a single multiplication there are several 'multiplications', the roleof which is played by the operations in Q. We note that the Q-semigroup(multioperator semigroup) introduced in the preceding section can also beregarded as a generalization of rings. Only here we have for one multipli-cation many 'additions'.From the definition it follows that an Q-group is an algebra relative to

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lattice of ideals of G as an Q-group. The kernel of this homomorphismconsists of the autornorph isms that leave every ideal o f G invariant.From the preceding remarks it also follows that all congruences of an

Q-group commute pairwise.

(1 )for all Xl' X2 E K, h I , 1 1 2 E H.When we apply the left distributive law, this inclusion can be rewrittenin the following way:

1.2.2. NEAR-RINGS. MULTIOPERATOR NEAR-RlNGS. In the preced ingsec tion we have defined fo r every Q-algebra G the Q-semigroup of all trans-formations of this algebra. If G is a group, this Q-semigroup is caned a near-ring. Near-rings are a special case of multi operator groups and with thispoint of view we now consider the ir simplest properties.

(2)Examining the case hI = 0 we conclude that xlh2 E H must necessarily

hold for all Xl E K and liz E H, tha t is, H must be a left ideal (in the senseof the theory of semigroups) in the multiplicative semigroup of K. Butnow (2) turns out to be equiva lent to the following condition:

Thus, a near-ring is a set K with two a lgebraic opera tions -- addition andmultipl icat ion, for which we make the following assumptions ;a) under addition K is a group (the additive group K+), but not nec-

essarily commutative;b) under multipl icat ion K i s a semigroup (the multipl icat ive semigroupK') ;c) multiplication and addition are linked by the left d istributive law

(3 )for any Xl' Xz E K and hE H.From all we have said it follows that a normal subgroup H of the additivegroup K + is an ideal if and only if H is a left ideal of the multiplicative

semigroup K' and if (3 ) holds.x(y + z)=xy + XZ. A near-ring K is said to be a quasi-ring if the group K+ is generated bydistributive elements. Just as above, it is easy to verify that in the defin ition

of a dis tributive element X i t i s suff icient to require dist ributivi ty relative toaddition: th is alone implies tha t fo r every y the equations (-- y) X =--yxand Ox= are valid.We now show that a normal subgroup H of the additive group of a quasi-ring K is an ideal in K if and only if H is a two-sided ideal of the multi-plicative semigroup K.

Strictly speak ing, we should have postulated tha t left distribu tiv ity a lsoholds for the remaining two operations of the additive group. However,we can show that this condition follows from the preceding one. Let X andy be arbit rary elements of a near- ring K and 0 the zero of the additive group.We establish the identities:X'O = 0; x(-y) = -xy.For, x (O + y) == X 0 + xy = xy, hence X 0 = .

Let H be an ideal of a quasi-ring K.Then by applying (3) to the case whenX2 is a distributive element we see that hxz E H. Now let X be an arbit raryelement of K. This x can be represented un the form x =:eiXj, whereeiXi = Xi and all the elements Xi are distributive. For every h E H we obtainhx = Ee.hx, EH, that is, H is a right ideal in K. The fact that H is a leftideal was established earlier.

Next, x(y - y) = 0 = xy + x(-y) and x(-y) = -xy.Now let us see what the definition of an ideal looks like in the case of near-

r ings. Let H be an ideal of a near-ring K. This means that H is a normalsubgroup of the additive group, and since in our case the system of opera tionsQ consists of the multiplica tion only, it remains to add the cond ition:

Now we assume that H is a two-sided ideal of the multiplicative semi-group K. We have to show that (3) holds.

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1 Algebraic systems 1.2 Multioperator groupsLet Xz =. :8;Y; , where al l the Yi are distributive elements. Then we have: and b). Finally, an Q-rin,~is an a-near-ring K such that under addition and

multiplication K is a ring, that K as an Q-group is Abelian (see 1.2.4) andthat multiplication is distributive relat ive to all operations of this Q-group.The simplest and most important example of such Q-rings are associativelinear algebras.It is easy to see that an ideal of an Q-near-ring K is a subset of K thatis an ideal of K as a near-ring and at the same time an ideal of K as an

Q-group .

- X,X2 + (x , + h) Xz =- X1X2 + 2,'eix1 + h) Yi == - XIXZ + D;;( .XIYi + hyJ iSince H is a normal subgroup of the additive group and all the h y; belongto H, we can write:._- X1XZ + (Xl + h) Xz =.c - X1Xz + l 'G;x1y, + h, =

i

Here h' EH, and consequently the required condition holds. We mentionthat according to the general Theorem J .6.3., which refers to Q-semigroups,every quasi-ring with a commutative additive group is a ring.After the preceding discussion it is natural to make the following general-

ization.Let G be an Q-group and If the semigroup of all transformations of G.We know already that the additive operat ions of G+ as well as all the multi-

operators of the system Q carryover immediately to the set 11. Here the zeroof H is the transformation that carries every element of the Q-group G intoits zero. From general arguments it follows that in this' way H also becomesan Q-group. Now we have in H the following three systems of operations:a) the operations of the additive group,b) the operations of Q,c) multiplication.By investigating the various combinations of these systems we come to

the following picture. Under a) and c) H is a near-ring, and under a) and b)H is an Q-group, as already mentioned. Hence it follows that H is a multi-operator group under the additive operation a) and the system of multi-operators consist ing of b) and c). Moreover, 11is a multi operator semigroupunder the multiplication c) and the system of multi operators consisting of a)and b).This leads us to the following definition.Every set K on which there are given systems of operations a), b), and c)

having the properties listed is called a multioperator (Q-) near-ring.We also define an O-quasi-ring as an Q-near-ring in which the correspon-

ding Q-group is generated by distributive elements. Of course, here we havein mind distributivity (multiplication) relative to operations of the type a)

1.2.3. NORMAL SERIES AND SYSTEMS. The concepts of a normal system,of an invariant system, of ascending and descending normal series play animportant role not only in the abstract theory of groups or the general caseof Q-groups, they are also fundamental for the discussion of representationsof groups by automorphisms of Q-groups. In their most general form theseconcepts were introd uced into the theory of groups in the survey paper [1 Jby Kurosh and Chernikov. The corresponding concepts for Q-groups differfrom the group-theoretical concepts only in that wherever normal subgroupsare mentioned we have to speak of ideals instead.We recall the definitions.Suppose that in an Q-group G a system of Q-subgroups [AaJ is given

containing the zero subgroup 0=Ao and the group G= itself, and thatthis system is ordered under set-theoretical inclusion. Such a system [A"Jis called com plete if the intersection and union of all the members of anysubsystem also belong to [AJ . A jump in [AaJ is a pair of members, sayA a and Ad 1, such that there are no otber members of the system betweenthem. If f A a J is a complete system, then every element g of G determines ajump: A a C g ) is the union of all members of the system that do not containg, and Ad l ( g ) the intersection of all members containing g. It is knownthat every system of Q-subgroups of a multi operator group that is orderedby inclusion can be completed.A complete system [ A a J is called normal if for every jump A a , Ad I the

Q-subgroup Aa is an ideal in Ax+ t- If all the Aa are ideals in G, then thesystem is called invariant. The Q-groups Aa+ 1 f A a connected with a normalsystem [ A a J are called the factors of the system.A normal system [A'a] id called a refinement of a normal system [AaJ

if [AJ i s a subsystem of [A' a J . If a normal system does not have non-tr ivialrefinements, it is called a composition system. Every Q-group has com-

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1 Algebraic systems 1.2 Multioperator groupsposition systems, and every normal system of such a group is containedin some composition system. One of the possible methods of classifyingQ-groups is based on properties of composition and other normal systems ofa given Q-group.For example, a system [A"J is called strongly soluble (soluble systems

will be defined later) if each of its factors is a commutative algebra. AnQ-group is called absolutely simple if i t has no non-t rivial normal systems.A simple Q-group, that is, one without non-trivial ideals, need 110t beabsolutely simple.A normal system is called an ascending (respectively, descending) nor-

mal series i f i t is asccndingly (respectively descendingly) well-ordered. AnQ-group is said to be strictly simply if it has no non-trivial ascending normalseries.In a recent paper [6J P. Hall has shown that there exist simple groups

that are not strictly simple. However, in [3J Levich has shown that in thecases of associative and Lie rings the concepts of simplicity and strictsimplicity coincide.Two normal systems are called isomorphic if a one-to-one correspondence

can be set up between their factors in such a way that corresponding factorsare isomorphic Q-groups. The classical Jordan-Holder theorem is valid forQ-groups: if an Q-group'has two fini te normal series 'with simple factors,then they are isomorphic. An analogon to this theorem for infinite series isthe following well-known theorem of Kurosh [2J:

1) the additive group G+ is commutative,2 ) all the operations of Q commute with the additive operations (it is

sufficient here to require permutabili ty with addit ion)3) G is commutative as an Q-algebra.A multioperator group in which only the first two condi tions are satisfied

is called Abelian.In this context we have to distinguish between two types of commutatorgroups. The commutator group of an Q-group G is the intersection of allideals of G having Abelian Q-factor groups. If G' is this commutator sub-group, then the Q-group GIG' is also Abelian.If we start out not from Abelianness but from commutativity, then we

come to the definition of the strong commutator subgroup of an Q-group G.This corresponds, obviously, to the commutator subgroup in the sense ofthe preceding section, when G is regarded as an algebra.Now we give another intrinsic definition of the commutator subgroup.Let A and B be two Q-subgroups of a multioperator group G. We define

the mutual commutator [A, B J of these two subgroup as the intersect ionof the ideals {A, B} that contain all the commutators[a, bJ = - a - b + a + b, a E A, b E B,

and all the elements of the form[aI' a2, , an; b1, bz, ... , bn; w] =- a1a2 anw -- b1b2 ... bnw ++ (a1 + bI)(a2 + b2) (an + bn)w,1.2.3.1. Any two ascending normal series with strictly simple factors of

an Q-group G are isomorphic. where w is an n-ary operation of n, a1> a2, , an EA andbI, b2, , b; EB. It is not hard to verify that [A, B J = [B, AJ.From the definition of the mutual commutator it follows immediatelythat if H is an ideal of G with an Abelian factor group GIH, then the mutual

commutator [G , GJ is contained in H . On the other hand, G/[G, GJ is ob-viously also an Abelian Q-group. Therefore the mutual commutator [G , GJcoincides with the commutator subgroup of the Q-group G.The concept of the mutual commutator has the following significance.

An Q-subgroup H of an Q-group G is an ideal in G if and only if the mutualcommutator [Fl, GJ belongs to H.From this remark it follows, in particular, that in an Abelian Q-group

every Q-subgroup is an ideal.Itnow remains to define the following concept.

The proof for multioperator groups is the same as that for groups withoutoperators, and is based enti rely on Zassenhaus' lemma. This lemma, as wellas the i somorphism theorem, remain val id for arbitrary Q-groups.In what follows we give for one special class of Q-groups an analogon to

Kurosh's theorem concerning series with simple factors (see 1.5.3.3) .

1.2.4. ABELJANNESS,NILPOTENCY,SOLUBILITY. In the case of Q-groupswith a non-trivial system of multioperators we have to distinguishbetween 'Abelianness' and 'commutativity'. An Q-group G is called com-mutative if it is commutative as an algebra, that is, if:24 25


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1 A lgebraic systems 1.2 Multioperator groupsA normal system [A~] of an Q-group G is said to be soluble if all its

factors are Abelian Q-groups. A system [A Jis said to be central if [Ao:+1, G]cAo: for every jump A~, Aa+ I' Starting out from these concepts wearrive at various classes of Q-groups of soluble or nilpotent type.Here soluble and nilpotent Q-groups are those that have, respectively, a

f inite soluble or central series.

of an Q-group G, and C is an ideal in A, then C is also an ideal in G. Wehave only to show that for every n-ary operation W EQ and arbitrarycl, Cz, ... , Cn E C and gl' g2' . , gn EG we have:

1.2.5. DIRECT DECOMPOSITIONS. COMPLETE REDUCIBILITY. The notionof the complete direct sum of algebras is applicable, in particular, to Q-groups. However, for multioperator groups we can also speak of directdecompositions, just as in the case of groups. A direct decomposit ion of anQ-group G is a direct decomposition of the additive group G+ in which allthe components are ideals.Now let [GJ, 0: E I,be a family of Q-groups and G their complete direct

sum. Then Gisa lso an Q-group. In G we consider the elements for which thecomponent with the suffix 0: ranges over the whole group G~, and all theremaining components are zero. The set of these elements, which we denoteby G~, is an ideal in G that is isomorphic to the Q-group Ga. We denote byG' the Q-subgroup of G generated by all these G~. This G' is called the(discrete) direct sum of the Q-groups G a o Thus, in the case of finitely manysummands the complete and the discrete direct sum are one and the samething. In the general case, the group G' splits into the direct sum of itsideals G~. This means that the operations of forming the direct sum anddirect decomposition can be regarded as inverses of one another.Many theorems on isomorphisms of direct decompositions carryover to

the theory of multioperator groups. We do not state these theorems here;we only mention that the problem of isomorphism of direct decompositionsalso has a close relationship with our theme.We now list certain facts on completely reducible Q-groups. An Q-group

G i s said to be com pletely reducible if each ideal A of it has a complementaryideal B such that G is the direct sum of A and B.

Every gi has the form gi = a, + bi, a, E A, b, E B, i = 1, 2, ... , n,Furthermore, from the definition of a direct decomposition it follows im-mediately that

Next we have:[ c l, Cb , cn; gl' gz, ... , gn; w ] = - C1Cz . .. CnW-- glgz g" w + (ci + gl)(CZ + gz) .. (cn + gn) W ==--CICz C"W -_ alaz ... anw - b1b2 .. bnw ++ (c1 + al)(cz + az) (cn + an) W + blb2 bnw ==cj, Cz, ... , Cll; ai' az, ... , an ; w ] E C,

from which it follows that C is an ideal in G.

1.2.5.1. Every ideal of a completely reducible Q-group is also completelyreducible.

Now let G be a completely reducible Q-group, A one of its ideals and Can ideal in A. Then C is an ideal in G and consequently has a complementaryideal D in G. Using the fact that the lattice of all ideals of an Q-group ismodular (it is a sublattice of the lattice or normal subgroups of the additivegroup G+) , we find that A = (C + D) nA =C + (D nA). The lemma isnow proved.From this lemma it also follows that a homomorphic image of a complete-ly reducible Q-group is also completely reducible. For by the isomorphismtheorem every homomorphic image of a completely reducible Q-subgroupis isomorphic to some ideal of the group, which by the lemma is completelyreducible.Now we can prove the following theorem (see, for example, Jacobson

[4]):We prove the following lemma:First we observe this property: If G=A + B is a direct decomposition 1.2.5.2. An Q-group is completely reducible if and only if it is a directsum of simple Q-groups.

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The next proposition follows from this theorem:

We split the set of ali direct summands into disjoint classes of pairwiseisomorphic ones and denote by Gp, f3 E1', the direct sum of the terms in oneclass. Then G=L. G p . We show that all these G li , which are called thehomogeneous components, do not depend on the method of splitting G intothe direct sum of simple summands.For example, let G p =LH y , ' ) ' E1", where all the H ; are isomorphic. Weshow that a simple ideal in G belongs to Gp if and only if it is isomorphic

to these H y -Let A be a simple ideal in G. It is easy to see that in the original de-

composition we can find finitely many terms whose sum contains A. Wedenote this sum by G' and let G p = G' (\ G p . It is clear that G' is the directsum of these Gfi . Now we assume that A is isomorphic to a component ofthe decomposit ion of Gp , but that A . Gp . This means that G' can be splitin two ways into the direct sum of simple Q-groups so that these decom-posit ions have distinct numbers of terms isomorphic to the simple groupsH y' ')' E1". The latter contradicts, for example, the Jordan-Holder theorem.Therefore , in this case A is contained in G J Conversely, suppose that an ideal A i s contained in Gli . We may assumethat A iscontained in Gli . Since A is a member of some direct decompositionof Gl i into the direct sum of simple terms, again by the Jordan-HOldertheorem (or by Schmidt 's theorem) A is isomorphic to all the f(" ')'EI",The invariance of the homogeneous components, which we mentioned

above, follows from this.We can also derive without difficulty the theorem that any two decom-

posi tions of an Q-group into' a direct sum of simple terms are i somorphic.We omit the proof of this theorem and consider the case when a com-

pletely reducible Q-group has a unique decomposition into the direct sumof simple terms.An ideal Z of an Q-group G is called central if [Z, GJ = O. The sum ofall central ideals of an arbit rary Q-group is also a central ideal and is calledthe centre of the Q-group (Higgins [lJ). A multi operator group is called agroup without centre if its centre is the zero subgroup.We show that if G is a completely reducible Q-group, then every Abelian

ideal in G is central.

Let G be a completely reducible Q-group. We show that it possesses simpleideals (that is, ideals that are simple as Q-groups). If G is simple, the pro-position is trivial. Suppose that G has a proper ideal A, and let a be anelement of G that does not belong to A. By Zorn's lemma we can find inG an ideal A' that is maximal relative to the property of containing A andnot containing a. Let B be a complement to A' in G. We show that B is asimple ideal. Suppose that this is not so. Then B as a completely reducibleQ-group splits into a non-trivial direct sum of ideals of G: B = BJ + Bz .Using the maximali ty of A' we now obtain a E (A' + B1) (\ (A' + B2) =A'.This is a contradiction, and therefore G

groups of auto morph isms of algebraic systems

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