THE

EASY GROUP THEORY

By Professor G. A. MILLER

UNIVERSITY OF ILLINOIS

HE report that the title of the chair of "differential and integral calculus" in the University of Paris has recently been changed to "the theory of groups and the calculus of variations" may tend to create a desire on the part of a larger group of scientific men to understand the essence of a mathematical group and the rôle which the group concept is assuming in modern mathematical developments. The fundamental importance of the concepts of differential and integral calculus in various fields of science has long been recognized, and the change of title noted above does not imply that the theory of groups and the calculus of variations tend to supplant the differential and integral calculus. It does, however, imply that the former subjects are also sufficiently fundamental and fruitful to merit prominent recognition in a leading mathematical center.

Paris may justly claim a very large share in the early development of group theory. A Parisian, A. L. Cauchy, is commonly regarded as the founder of this subject, while other Parisians, including A. T. Vandermonde and E. Galois, did important pioneering work in this field. Although J. L. Lagrange is commonly regarded as a French mathematician his pioneering work in group theory was done before he settled in Paris. The first separate treatise on this subject was written by a Parisian. This work appeared in 1870 under the title Traité des substitutions et des équations algébriques by C. Jordan.

In group theory as well as in differential and integral calculus the first extensive formal or abstract developments are due to English and German mathematicians. In the case of the latter subjects the authors of these developments, viz., Newton and Leibniz, are commonly regarded as its founders. Fortunately this has not yet been done as regards the former subject. Notwithstanding the fundamental importance of formal developments and abstract formulations the real life of mathematics abides in its contact with the concrete. In the case of group theory this contact has been emphasized especially by S. Lie, F. Klein and H. Poincaré. The rapidly increasing prominence of this theory during the last half

century is largely due to the writings of these three men, who have been leaders also in various other lines of mathematical activity.

A dominating mathematical concept, like a dominating personality, has a charm of its own, and creates a kind of atmosphere which is as invigorating as that of a real university or that which emanates from any group of real scholars. A fundamental notion of a mathematical group is that "there is no new thing under the sun." It is true that we speak here of generators and generational relations, but the objects which are generated were members of the group since the beginning of time and will remain members thereof throughout eternity. We study the group to perceive this sameness in its various forms and to understand the relative properties of the various elements which unite to make a single element of the group.

The non-technical meaning of the term group suggests little as regards its technical meaning except the invariance of the number of its elements. In a non-technical group one usually thinks of the elements as units which may or may not have the power of reproduction. In the former case the elements thus produced are usually new elements of the group. In the technical group the elements have necessarily the power of uniting, but when they unite they neither produce any new element nor lose their own identity. The union merely exhibits the possible decomposition of an element of the group, or the different ways of securing e pluribus unum. Union, unity and stability constitute the triumvirate of the theory of groups. The stability here noted is not the stability of statics but the stability of dynamics. It is a kind of invariance under transformations.

What is perhaps of most interest in this connection as regards the non-mathematician is the question why the concept to which we referred above is so fundamental in mathematics. It is well known that mathematical developments have been largely guided by the desire to secure intellectual penetration into the workings of nature. Do we find in nature numerous instances of the union of elements of the same kind to produce an element of this kind which is really not new but belongs to a totality which has been clearly defined? For instance, one may think of the totality of the transformations of space subject to the condition that the distance between every pair of points in space remains invariant. It is evident that if one such transformation is followed by another the two together are equivalent to a single transformation of the totality in question. Hence we say that this totality constitutes a group.

Vol. XV.-33.

It is also clear that the totality of the natural numbers when they are combined by addition has the property that no new number arises from the combination of any two of them, or from the combination of one with itself. In both this totality and in the totality of transformation noted in the preceding paragraph it is evident that if x, y, z represent three elements of the totality and if any two of them are supposed to be known then the third is completely determined by the following equation:

xy=2

The term group is commonly used in mathematics in the restricted sense that the third of any three elements is completely determined by such an equation where any two of them are supposed to be given. Moreover, it is usually assumed that when any three elements are combined the associate law is satisfied, but it is not assumed that the commutative law is necessarily satisfied when two of them are combined.

Even when these restrictions are imposed there are instances of groups almost wherever one turns. It is true that in the vegetable and the animal kingdoms one sees new elements arising in profusion, and mathematics is naturally called on to deal also with such conditions, but if one looks deep enough here there seems to be a union of elements without loss of identity of the elements, and there seems to be nothing new in the profound formal physical sense. Hence one may see some significance in the following statement made by Poincaré shortly before his death: "The theory of groups is, so to say, entire mathematics, divested of its matter and reduced to a pure form."

In the groups noted above the number of elements is infinite. As instances of a finite group we may consider the six movements which transform a fixed equilateral triangle into itself and the eight movements which transform a fixed square into itself. Such special instances are, of course, of little interest to the general scientist except as illustrations. The thing that may be supposed to command the interest even of the educated layman is the fact that these very evident considerations appertain to a profound mathematical theory.

Group theory did not arise from such obvious considerations. After it was partially developed as an autonomous science some of its more obvious applications received special attention. A certain degree of difficulty seems to create the most favorable atmosphere for scientific developments. In group theory this atmosphere was created by the n roots of the algebraic equation of the following form:

It is customary to speak of x as the unknown and to call this an equation of degree n in one unknown. As a matter of fact the equation has n roots and all of these are unknown, so that it is really an equation in n unknowns. The constant coefficients a1, a2 an were known to be symmetric functions of these n unknowns before the subject of group theory was developed.

The mysterious appearance of n unknowns to take the place of the one which presents itself openly is perhaps sufficient to create. an atmosphere suitable for scientific endeavors. At any rate, it was in this atmosphere that our subject arose and hence we shall note here a few of the early steps in its development. It is true that these steps were taken by men who seemed to have no idea that they were dealing with notions which had the widest applications in other fields of mathematical endeavor. In fact, the early explorers of group theory died long before any one realized that the notions with which they were dealing were destined to permeate a large part of the science of mathematics.

The n roots of an equation of degree n with constant coefficients constitute a group in the non-technical sense of the term, but the group of this equation is something very different and lies much. deeper. It relates to a certain totality of permutations of these n roots, or substitutions on these n roots, leaving invariant every possible rational function of these roots which is equal to a constant, and having the property that if a rational function of these roots is invariant under these substitutions it is equal to a constant. These fundamental properties of the n roots of the equation in question were first noted by E. Galois, a French mathematician of great renown, although he died before reaching the age of twenty-one years. They were sufficiently difficult to create an atmosphere suitable for the development of our subject as an autonomous science.

A study of the permutations of n things might at first appear to promise little of importance. It is true that before the time of Galois attention had been called to this subject by Lagrange and Vandermonde in connection with the question under what condition a rational function on n variables can be expressed rationally in terms of another such function, but it was not until long after the days of Galois that mathematicians began to realize the fundamental importance of this subject in the study of a large variety of mathematical questions. When the substitutions arising from these various permutations were studied by themselves they were seen to combine according to laws which are found almost everywhere when the data are sufficiently connected to admit mathematical treatment.

The reader who has given little attention to mathematical developments may be inclined to ask, If the notion group is so fundamental why did the ancients and even such eminent later thinkers as Descartes and Newton pay no explicit attention to it? Why does the theory of groups not have an ancient prototype like differential and integral calculus, whose prototype is found in the method of exhaustion of the ancient Greeks? Does it appear reasonable that a subject founded only three quarters of a century ago should really deserve such a dominating position in modern. mathematics as is claimed for group theory in what precedes the present paragraph? Does the history of mathematics present any other instance of the sudden rise of a dominating concept extending into practically all the large branches of mathematics?

The fact that the last of these questions must be answered in the negative tends to enhance the interest in the others. This negative answer calls also for a word of caution for there is danger that the reader might infer from it that the subject of group theory has greater merit than really belongs to it. It seems that mathematical developments have always been guided by the group concept. In the words of Poincaré "the ancient mathematicians employed groups in many cases without knowing it." The main question in mathematical developments is that one gets on the right road. The ability to explain why one has chosen this road is of secondary importance. In fact, the best teacher is frequently unable to give a good account of his methods while a much inferior teacher may be able to talk glibly about methods.

Group theory is largely a method and those who are studying this subject by itself may be compared with those who are devoting their attention to methods of teaching. Just as the latter are not necessarily the best teachers so the former are not necessarily the best mathematical investigators. Possibly the bacteria which have tended to make the teachers colleges such a prominent feature of our modern universities have also caused the emphasis on group theory in modern mathematics. Just as some of our best teachers have never read a work on methods of teaching so some of the best mathematical investigators have never secured a speaking acquaintance with the notion of group. In both cases the real essence of the subject has been acquired unconsciously, or, at least, without the development of a formal language relating thereto.

The fact that modern mathematicians emphasize the group concept while the ancient and medieval mathematicians did not do this does not imply a change of mental attitude. Naturally the modern mathematicians secured a somewhat deeper insight into various subjects and thus discovered evidences of groups