Table Of ContentALGEBRA
Volume 1
BY
L. REDEI
Mathematical Institute, University of Szeged
Szeged, Hungary
PERGAMON PRESS
OXFORD . LONDON - EDINBURGH . NEW YORK
TORONTO • SYDNEY • PARIS . BRAUNSCHWEIG
Pergamon Press Ltd., Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W. 1.
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Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig
Copyright © 1967
AKADEMIAI KIAD6, HUNGARY
First English edition 1967
This is a translation of the original
Hungarian Algebra published
by Akademiai Kiado, Budapest
Library of Congress Catalog Card No. 64-24548
2016/67
PREFACE TO THE GERMAN EDITION
Algebra originated from the connections performable in sets of natural
numbers. This surely constitutes one of the oldest collective accomplishments
of mankind. With the development of civilization numerous other examples
of connections came into being before it was realized that they could be
viewed from a uniform standpoint and that far-reaching generalizations
could be reached, i.e. in each set one or more connections can be defined
by aattching one element of it to each two elements of the same set, where
after it can be termed a structure. Algebra deals with the investigations
into such structures.
Algebraic research is limited by restrictions. One of these is that of all
imaginable connections only those of practical importance must be investig
ated. Such are associative connections wherefore the author has confined
himself to this case only, as is generally done in text-books. Non-associative
connections are merely considered as expedients.
What is of primary importance is the modification of that notion of Al
gebra which springs from the realization that the nature of the elements of a
structure need not be considered. The exact formulation of this idea is owed
to the genius of Ernst Steinitz who by establishing the "Principle of Iso-
morphy" in his epoch-making treatise "Algebraic Theory of Bodies"
created therewith something similar in algebra to what Felix Klein had done
in geometry in his "Programme of Erlangen".
As algebra is a relatively young branch of science, one should not be
surprised at the tremendous amount of algebraic studies in recent years.
Though many of its chapters were extensive before Steinitz, development
remained incomplete and its parts were not sufficiently coordinated with
one another as a consequence of the lack of a uniform standpoint, and at
present it still falls behind other mathematical disciplines in completeness
and method. This suggests the necessity for continued research.
The author believes that a methodical approach is to set out from gene
rality and to proceed to the particular. A systematization of this kind was
attempted but such efforts had to be kept within bounds, as the author want
ed to write a text-book. These requirements, partly contradictory, have been
met by placing the general formation of concepts at the beginning of Chap
ters II and III, and by dealing with special cases of application either parallel
to the exposition or immediately following it. In these two chapters some
problems of structure are treated to satisfy the natural demands of students
xi
xii PREFACE TO THE GERMAN EDITION
who are mostly interested in the elaboration of structures. In order to
prevent any further extension of Chapters II and III, which are already
a little too comprehensive, applications had to be used sparingly, but further
applications are dealt with in Chapters IV—XII which are devoted to various
other topics.
In accordance with the author's efforts at systematization much attention
has been paid to the analogies between group theory and ring theory; it
would be very desirable to find more such analogies.
Among other text-books not mentioned individually, the author has
drawn largely from B. L. van der Waerden's "Algebra" and Pickert's
"Introduction to Advanced Algebra". It must be mentioned that R. Kochen-
dorffer's recently published "Introduction to Algebra" has enabled the
author to bridge some gaps in his completed manuscript.
The contents of paragraphs 52, 53, 54, 85, 92, 94, 95, 120, 122, 123, 146,
178,180,183,184,185 have not previously been dealt with in any text-books
and are, for the most part, completely new. The same is largely true for
paragraphs 27, 44, 49, 51, 58, 59, 64, 65, 66, 73, 82, 103, 121, 134, 148, 172,
173; in these the novelty consists merely of the fact that well-known theo
rems have been worded in a more general way than hitherto. More or less
new methods are contained in paragraphs 11, 33, 50, 89, 91, 106, 115, 130,
136, 138, 147, 153, 169, 170, 179.
The reader of this book requires little mathematical knowledge beyond a
knowledge of natural numbers; at the very most a certain aptitude for
mathematical thought is required. Fundamental questions have not been
taken into consideration by the author.
Examples, exercises and some unsolved problems are to be found at the
end of many paragraphs. Some examples have also been inserted in the
text itself where they are necessary for a better understanding of the subject-
matter treated. In the exercises and examples some well-known concepts
(e.g. that of complex numbers) have been assumed; these are defined in a
subsequent part of the book. The reader is advised to study the examples
carefully. Paragraphs marked with an asterisk require considerable effort.
The reader may omit the proofs at a first reading.
For the sake of simplification the customary terminology has been
changed slightly. The author says, e.g. (analogously to "coefficient group")
"coefficient ring" instead of "remainder-class-ring". The "full permutation
group" (quite analogous to the "full matrix-ring") means the "symmetrical
permutation group". A "main-polynomial" means a polynomial with the
initial coefficient of 1. Instead of using the usual "main-polynomial of an
element" the author uses "minimal polynomial of an element". Though
the author talks of "normal or standardized polynomials" in the customary
sense, the notion of the "main-polynomial", however, can still not be
avoided.
PREFACE TO THE GERMAN EDITION xiii
It was the author's endeavour to use as simple notations as possible.
As a novelty in this field of notation, the system S ~ S' (a -> a') should be
mentioned, by which the author wishes to express that a -> a' is a homo-
morphic transformation of the structure S on structure S'.
The author has made great efforts to facilitate the reader's task. This
purpose is served by cross-references to serial-numbered theorems at the
bottom right-hand side of odd-numbered pages.
In spite of the size of this volume completeness in any direction could not
be hoped for. In particular, in the theory of finite groups some interesting
special themes have been treated, but only a few fundamental theorems
are given. It would have been desirable to include "General Algebra"
which is becoming more and more important but this had to be renounced
owing to pressure of space.
Besides some text-books and monographs only such other works are
included in the bibliography as have been quoted in the text.
Many modifications have taken place since the first Hungarian edition
of this volume. Some paragraphs (mainly in Chapters II and III) have been
rearranged, some have been split, and further on many new paragraphs have
been added. Some errors to which the attention of the author was drawn,
partly by his colleagues and readers, have been corrected. For this and many
other services of various kinds the author owes a great debt of gratitude
to Messrs. A. Adam, W. Blaschke, G. Fodor, E. Fried, L. Fuchs, L. Kalmar,
F. von Krbek, G. Pickert, G. Pollak, L. Prohaska, L. Pukanszky, S.
Schwarz, E. Sperner, O. Sternfeld, J. Szendrei. The author is deeply grateful
to his late friend, T. Szele, who was of great assistance.
The author's sincere acknowledgements are due to the Academic Publish
ing Company Geest&Portig K.-G., Leipzig, and the printing office
"Magnus Poser" at Jena, who have always willingly met numerous special
wishes, often with difficulty.
After the delivery of the manuscript in April 1956, various changes were
made, so that this work was concluded only at the end of 1958.
Szeged, December, 1958 The Author
PREFACE TO THE ENGLISH EDITION
Compared with the original German edition this volume contains the
results of more recent research which have to some extent originated
from problems raised in the previous German edition. Moreover, many
minor and some important modifications have been carried out. For
example paragraphs 2 — 5 were amended and their order changed. On the
advice of G. Pickert, paragraph 7 has been thoroughly revised. Many
improvements originate from H. J. Weinert who, by enlisting the services
of a working team of the Teachers' Training College of Potsdam, has sub
jected large parts of this book to an exact and constructive review. This
applies particularly to paragraphs 9, 50, 51, 60, 63, 66, 79, 92, 94, 97 and
100 and to the exercises. In this connection paragraphs 64 and 79 have had
to be partly rewritten in consequence of the correction.
Besides those already mentioned the author wishes to express his thanks
to his colleagues for their comments and advice and to the publishers and
printers for their careful work.
Szeged, July, 1964 The Author
XV
LIST OF SYMBOLS
€ 1 *() 48
0 1 V 49
<> 1
c 2 aI| /S 50
U 2
n 2 a 57
— 2 — a 57
X 2,252 # a 57
-+ 4 0+() 59
("1 5 £ 58
k
/t! 61
a-1 6,30 R* 64
r 9 <J>~) 66
r^ 10 F* 67
<o 10 Q 68
= 10,154 ',7, & 69,300
< 19,568 {} 76
min, max 19 0 () 81
o 28 Z() 82
+ 28 & 87
• 28 S K S' (a -> a') 88
0 32,43 /^/ 92
s+ 37 S ~ S' (a -> a') 92
sx 37 s/e 98,126,129
o() 38 mod 98,154
00 38 3ft 111
,4-Structure 38 *() 113
S|<9 40,226 S° 117
(9-Structure 40 O (G : H) 122
fo 40 () 130
Jf 40 O* 130
<JT 40 0, 130
n<^ r, o> 40 sign 135
42,592 ■*. 136
F* 46 ® 141
a" 47 e 141
xvii
xviu LIST OF SYMBOLS
i°G 175 .«() 382,384
s- 182 o (a (mod m)) 393
J? + R 188
, d 0
f 434-6
A, V 214 ' ' dx'' dx
inf, sup 216 a*|||8 440
J?- 230,231,232 Rif,9) 444
F|^ 232 D{f) 451
S | 0 ~ S' | (9 (a -* a') 224 477
^P> F,
<? 242
F | F x F | JF (a -> a') 479
Jf(H) 255
F() 482
J?[] 259
[G:F] 482
^Mo 263
r,r* 509
*•.(*)
264
F-" 542,547
(°Xn, (««) 269
551
^„ 274 /G|F, ^G|F , ?G|F
3/ 275 5G|F> ^GIF 556
275 <X), ^0 557
^
282 /.(*) 558
l«,.l
1 1 583
A 289
adjf 289 coO 586
(p-\im 587
NO, S() 296,551
602
i 297 ^0)
log 609
Q 298
n(8 ), *() 301 ^0) (0 610
ass 306 <^0 613
o;> []; 308 ^, 614
o*, [i* 310 H-O 620
a° 325 >M) 630
<K) 333,585 «^(p) 632
Q« 345 (fl 665
[3 366
(e,*) 673
CHAPTER I
SET-THEORETICAL PRELIMINARIES
§ 1. Sets
The notion of a set, i.e. that of a collection of certain various entities, is
taken to be known. A fuller treatment of set theory may be found elsewhere,
e.g. in FRAENKEL (1946). The entities of which a set consists are called the
elements of the set."We denote that a is an element of the set S as follows:
a£S
a$ S means that a is not an element of S. Later we similarly denote by putting
a stroke through a sign in this way that the assertion, which would otherwise have
been expressed by the sign, is false.
The notion of the empty set, denoted by O, is also useful. This is defined
as the "set with no elements".
If we give up the restriction that the elements are all different, then we
use the term system (instead of set). (For further details see the end of § 8.)
Two sets are called equal, when each contains the elements of the other.
We denote by <a, b,.. .> the set consisting of the elements a,b,...
Hence we have, for instance, <#, b, c} = (b, a, c>. ("=" is the sign used to
denote equality.) We often identify the set <#>, consisting of a single element
a, with the element a,
We often define a set by some property of its elements, e.g., the set of
unmarried men in London who are at least thirty years old. If no element
satisfies the defining property, then the set so defined is the empty set.
We can also think of sets as entities. Thus we allow the existence of sets,
whose elements are also sets. If, for example, A, B,.. . are different sets
then S = <v4, B,. . .> denotes the set with elements A, B,.. . Such a set S
is called a set of sets. Similarly we can have the set «a, b, c>, <b, d), (a, b}}.
If we have two sets S and T7, we say that T is a subset of S, and we write
if S contains all the elements of T. Alternatively we say that S is an overset
or extension set of T. Further if T<^= S and T ^ S, we call T a proper subset of S
and 5* a proper overset or proper extension set of T, and we write TczS.
1
2 SET-THEORETICAL PRELIMINARIES
It is reasonable to take A => B to mean the same as B c A. For other signs, to
be introduced later, we shall interpret oppositely directed signs in a similar way.
Given sets S,T,. . .,
their union, denoted by S U rU . . .,
and their intersection (or cross-cut), denoted by S f) T D . . .,
are taken to mean, respectively, the set of all elements occurring in at least
one of the given sets and the set of all elements occurring in each one of the
given sets.
By the difference set A — B of two sets A, B we mean the set of all elements
of A not contained in B. The sets A, B are called disjoint if A f] B = O.
The following abbreviations are sometimes used:
U A = A \J ... UA„
i 1
1=1
n A = A n ... n A .
t X n
1=1
Given a collection of sets, we say one of them is minimal or maximal, if
it is not a proper overset or not a proper subset, respectively, of any one of
the given collection of sets.
For example, among the sets <a, b, c>, <c, d) and <y>, the first and third are mini
mal, the first and second are maximal. Among the sets
</!,«+ 1,...> (n = 0, ± 1,± 2, ...)
there is neither a minimal nor a maximal set.
By the product of two sets A, 1? we mean the set of ordered pairs
(a, b) (aeA,be B)
and denote it by A x B. Here we have to point out that in general two such
ordered pairs, (a, b) and (c, d), are considered equal only when a = c and
b = d. The definition of a product is extended in the obvious way to the
product AxBx... of several sets A, B, . ..
When S = S = . . . = S , the product S x S x. .. x S is written S"*
± 2 n x 2 n
and called the nth power of the set S (n = 1,2,.. .).
It is well known that the so-called naive set-concept, applied above, involves
contradictions. This was realized by CANTOR the founder of set theory. However
these contradictions may be avoided by means of an axiomatical establishment of
set theory, which implies a restriction of the general set-concept. (See e.g., FRAENKEL
(1946), or, in more modern treatment, G6DEL(1953). We shall consider only such
sets as conform to the established principle of the limited axiomatical set-concept.
