Popular Science Monthly/Volume 64/February 1904/What Is Group Theory?

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By Professor G. A. MILLER,


IN the recent International Catalogue of Scientific Literature, group theory is classed among the fundamental notions of mathematics. The two other subjects which are classed under this heading are 'foundations of arithmetic' and 'universal algebra.' While it might be futile to attempt to popularize those recent advances in mathematics which are based upon a long series of abstract concepts, it does not appear so hopeless to give a popular exposition of fundamental notions. In what follows we shall aim to give such an exposition of some of the notions involved in the theory of groups.

This theory seems to have a special claim on popular appreciation in our country because it is one of the very few subjects of pure mathematics in whose development America has taken a prominent part. The activity of American mathematicians along this line is mainly due to the teachings of Klein and Lie at the universities of Göttingen and Leipzig respectively. During the Chicago exposition, the former held a colloquium at Evanston, in which the fundamental importance of the subject was emphasized and thus brought still more prominently before the American mathematicians.

There is probably no other modern field of mathematics of which so many prominent mathematicians have spoken in such high terms during the last decade. In support of this strong statement we quote the following:

There are two subjects which have become especially important for the latest development of algebra; that is, on the one hand, the ever more dominating theory of groups whose systematizing and clarifying influence can be felt everywhere, and then the deep penetrations of number theory.[1] The theory of groups, which is making itself felt in nearly every part of higher mathematics, occupies the foremost place among the auxiliary theories which are employed in the most recent function theory.[2] In fine, the principal foundation of Euclid's demonstrations is really the existence of the group and its properties. Unquestionably he appeals to other axioms which it is more difficult to refer to the notion of group. An axiom of this kind is that which some geometers employ when they define a straight line as the shortest distance between two points. But it is precisely such axioms that Euclid enunciates. The others, which are more directly associated with the idea of displacement and with the idea of groups, are the very ones which he implicitly admits and which he does not deem even necessary to enunciate. This is tantamount to saying that the former are the fruit of later experience, that the others were first assimilated by us, and that consequently the notion of group existed prior to all others. . . . What we call geometry is nothing but the study of formal properties of a certain continuous group, so that we may say space is a group.[3]

From these words of Poincarế it follows that the group concept is implicitly involved in some of the earliest mathematical developments. In an explicit form it first appears in the writings of Lagrange and Vandermonde in 1770. These men inaugurated a classic period in the theory of algebraic equations by considering the number of values which a rational integral function assumes when its elements are permuted in every possible manner. For instance, if the elements of the expression are permuted in every possible manner, it will always assume one of the following three values: , , .

The eight different permutations which do not change the value of one of these expressions are said to form a permutation group and the expression is said to belong to this group. There is always an infinite number of distinct expressions which belong to the same permutation group. Hence it is convenient for many purposes to deal with the permutation group rather than with the expressions themselves. This fact was recognized very early and led to the study of permutation groups, especially in connection with the theory of algebraic equations. The most fundamental work along this line was done by Galois, who influenced the later development most powerfully, although he died when only twenty years old.

Galois first proved (about 1830) that the solution of any given algebraic equation depends upon the structure of the permutation group to which the equation belongs. As the algebraic solution of equations occupies such a prominent place in the history of mathematics this discovery of Galois furnished a powerful incentive for the study of permutation groups. Before Galois an Italian named Ruffini and a Norwegian named Abel had employed permutation groups to prove that the general equation of the fifth degree can not be solved by successive extraction of roots. In doing this the former studied a number of properties of permutation groups and is therefore generally regarded as the founder of this theory.

The definition of a permutation group is very simple. It is merely the totality of distinct permutations which do not change the formal value of a given expression. Such a totality of permutations has many remarkable properties. One of the most important of these is the fact that any two of them are equivalent to some one. That is, if any one of these permutations is repeated, or is followed by some other permutation in the totality, the result is equivalent to a single permutation in the totality. This property is characteristic; for if any set of distinct permutations possesses this property they form a permutation group and it is possible to construct an infinite number of expressions such that they are unchanged by these permutations but by no others.

Soon after the fundamental properties of permutation groups became known, it was observed that many other operations possess the same properties. This gradually led to more abstract definitions of the term group. According to the earliest of these any set of distinct operations such that no additional operation is obtained by repeating one of them or combining any two of them was called a group. All the later definitions included this property, but they generally add other conditions. These additional conditions are frequently satisfied by the nature of the operations which are under consideration and hence do not always require attention. This may account for the fact that the oldest definition is still very commonly met in text-books, notwithstanding the fact that the ablest writers on the subject abandoned it a long time ago.

The three additional conditions which a set of distinct operations must satisfy in order that it becomes a group when the operations are combined are: (1) The associative law must be satisfied; i. e., if represent any three operations of the set, then the three successive operations must give the same result independently of the fact whether we replace or by a single operation. The operations are, however, not generally commutative, that is, may be different from . (2) From each of the two equations , it follows that . (3) If the equation involves two operations of the set the third element of the equation must also represent an operation in the set. It may be observed that the totality of integers combined by multiplication obey all these conditions except the last. Hence this totality does not form a group with respect to multiplication, although the contrary has frequently been affirmed.[4]

One of the simplest instances of a group of operations is furnished by the different numbers which satisfy the equation . It is very easy to see that these numbers obey each of the four given conditions when they are combined by multiplication. Hence we say that the roots of the equation form a group with respect to multiplication. Since all these roots are powers of a single one of them this group is said to be cyclic. Cyclic groups are the simplest possible groups and they are the only ones whose operations can be completely represented by complex numbers.

Another very simple category of groups of operations is furnished by the totality of movements which leave a regular polygon unchanged. For instance, a regular triangle is transformed into itself when its plane is rotated around its center through 120° or through 240°. Moreover, its plane may be rotated through 180° around any of its three perpendiculars without affecting the triangle as a whole. These five rotations together with the one which leaves everything unchanged (known as the identity) are all the possible movements of the plane which transform the given triangle into itself. Hence these six movements form a group, which happens to be identical with the group formed by the six possible permutations of three things.

It is not difficult to see that a plane can have just eight movements which do not affect the location of a given square in it. These consist of the three movements around the center of the square through 90°, 180° and 270° respectively; the four movements through 180° around the diagonals and the lines joining the middle points of opposite sides; and the identity. This group of eight operations has exactly the same properties as the permutation group on four letters which transforms into itself. Hence these two groups are said to be simply isomorphic. From the standpoint of abstract groups, such groups are said to be identical.

In general, a regular polygon of n sides is left unchanged as a whole by just movements of its plane, viz., movements around its center and rotations through 180° around its lines of symmetry, in addition to the identity. The first movements together with the identity clearly form a group by themselves. Such a group within a group is known as a subgroup. This category of groups of operations is known as the system of dihedral rotation groups or the system of the regular polygon groups. It is not difficult to prove that each of them is generated by some two non-commutative rotations through 180° and that no other groups have this property.

Among the non-regular polygons the rectangle with unequal sides has perhaps the most important group. There are clearly just three movements of the plane (besides the identity) which transform such a rectangle into itself, viz., the rotation through 180° around the center and the rotation around its two lines of symmetry through the same angle. These four operations form a group which presents itself in very many problems and is known by a number of different names. Among these are the following: four-group, anharmonic ratio group, axial group, quadratic group, rectangle group, etc. Since we arrive at the identity by repeating any one of its operations, it is entirely different from the group formed by the four roots of the equation . It is easy to prove that these two groups represent all the possible types of groups of four operations; that is, there are only two abstract groups of four operations. In general, the number of groups which can be formed with operations increases very rapidly with the number of factors of . When or 12 the number of possible abstract groups is 5.

Similarly, all the movements of space which transform a given solid into itself form a group. For instance, the cube is transformed into itself by twenty-four distinct movements. Nine around the lines which join the middle points of opposite faces, six around those which join the middle points of opposite edges, eight around the diagonals, and the identity. The group formed by these twenty-four movements is simply isomorphic with the one formed by the total number of permutations of four things. The regular octahedron has the same group, while the group of the regular tetrahedron is a subgroup of this group. The icosahedron and the duodecahedron have a common group of sixty operations. The groups of the regular solids play an important role in the theory of transformations of space. They are treated at considerable length in Klein's 'Ikosaeder' as well as in many other works.

All the preceding examples relate to groups of a finite number of operations, or of a finite order. During recent years the applications of groups of infinite order have been studied very extensively. As the theory of groups of finite order had its origin in the theory of algebraic equations, so the theory of groups of an infinite order might be said to have had its origin in the theory of differential equations. The rapid development of both of these theories is, however, due to the fact that much wider applications soon presented themselves. This is especially true of the latter. In fact, the earliest developments of the groups of infinite order were made without any view to their application to differential equations.

One of the simplest examples of groups of infinite order is furnished by the integral numbers when they are combined with respect to addition. The totality of the rational numbers clearly becomes a group when they are combined with respect to either of the operations addition or multiplication. The same remark applies evidently to all the real numbers as well as to all the complex numbers. These additive groups of infinite order are frequently represented by the equation , where may assume all the values of one of the given groups. If a may assume all real values the group is said to be continuous. When is restricted to rational values the group is said to be discontinuous, notwithstanding the fact that it transforms every finite point into a point which is indefinitely close to it.

While these examples exhibit a very close relation between continuous and discontinuous groups of infinite order, yet the methods employed to investigate problems belonging to these groups are generally quite different. The theory of the former is mainly due to Sophus Lie and has been developed principally with a view to the solution of differential equations. The theory of the latter has been developed largely in connection with questions in function theory and owes its rapid growth to the influence of Klein. A large part of Lie's results are contained in his 'Transformationsgruppen,' consisting of three large volumes, while the 'Modulfunctionen' and 'Automorphe Functionen' of Klein and Fricke are the best works on the discontinuous groups of infinite order.

Although the notion of group is one of the most fundamental ones in mathematics, yet it is one which is more useful to arrive at reasons for certain results and at connections between apparently widely separated developments than to furnish methods for attaining these results or developments. Its greatest service so far has been its unifying influence and its usefulness in proving the possibility or the impossibility of certain operations. In fact, it is generally conceded that group theory had its origin in the use which Ruffini and Abel made of it to prove that the general equation of the fifth degree can not be solved by radicals.

While it may be said to have 'shown its dominating influence in nearly all parts of mathematics, not only in recent theories, but also far towards the foundation of the subject, so that this theory can no longer be omitted in the elementary text-books,'[5] yet this influence is largely a guiding influence. The bulk of mathematics is not group theory and the main part of the work must always be accomplished by methods to which this notion is foreign. On the other hand, it seems safe to say that this theory is not a fad which will pass into oblivion as rapidly as it rose into prominence. Its applications are so extensive and useful that it must always receive considerable attention. Moreover, it presents so many difficulties that it will doubtless offer rich results to the investigator for a long time.

  1. Weber, 'Lehrbuch der Algebra,' vol. 1, 1898, preface.
  2. Fricke und Klein, 'Automorphe Functionen,' vol. 1, 1897, p. 1.
  3. Poincaré, The Monist, vol. 9, 1898, pp. 34 and 41.
  4. Among other places this error is found in the first edition of Weber's classic work on algebra, vol. 2, p. 54. It has been corrected in the second edition of this work. Somewhat simpler definitions of the term group have recently been given by Huntington and Moore, Bulletin of the American Mathematical Society, vol. 8, p. 388.
  5. Pund, 'Algebra mit Einschluss der elementaren Zahlentheorie,' 1899, preface.