Part 4

The most important of the applications of continuous groups are to the theory of systems of differential equations, both ordinary and partial; in fact, Lie states that it was with a view to systematizing and advancing the general theory of differential equations that he was led to the development of the theory of continuous groups. It is quite impossible here to give any account of all that Lie and his followers have done in this direction. An entirely new mode of regarding the problem of the integration of a differential equation has been opened up, and in the classification that arises from it all those apparently isolated types of equations which in the older sense are said to be integrable take their proper place. It may, for instance, be mentioned that the question as to whether Monge's method will apply to the integration of a partial differential equation of the second order is shown to depend on whether or not a contact-transformation can be found which will reduce the equation to either [Pd]^2z/[Pd]x^2 = 0 or [Pd]^2z/[Pd]x[Pd]y = 0. It is in this direction that further advance in the theory of partial differential equations must be looked for. Lastly, it may be remarked that one of the most thorough discussions of the axioms of geometry hitherto undertaken is founded entirely upon the theory of continuous groups.

_Discontinuous Groups._

We go on now to the consideration of discontinuous groups. Although groups of finite order are necessarily contained under this general head, it is convenient for many reasons to deal with them separately, and it will therefore be assumed in the present section that the number of operations in the group is not finite. Many large classes of discontinuous groups have formed the subject of detailed investigation, but a general formal theory of discontinuous groups can hardly be said to exist as yet. It will thus be obvious that in considering discontinuous groups it is necessary to proceed on different lines from those followed with continuous groups, and in fact to deal with the subject almost entirely by way of example.

Generating operations.

The consideration of a discontinuous group as arising from a set of independent generating operations suggests a purely abstract point of view in which any two simply isomorphic groups are indistinguishable. The number of generating operations may be either finite or infinite, but the former case alone will be here considered. Suppose then that S1, S2, ..., S_n is a set of independent operations from which a group G is generated. The general operation of the group will be represented by the symbol S_a^[alpha]S_b^[beta] ... S_d^[delta], or [Sigma], where a, b, ..., d are chosen from 1, 2, ..., n, and [alpha], [beta], ..., [delta] are any positive or negative integers. It may be assumed that no two successive suffixes in [Sigma] are the same, for if b = a, then S_a^[alpha]S_b^[beta] may be replaced by S_a^([alpha] +[beta]). If there are no relations connecting the generating operations and the identical operation, every distinct symbol [Sigma] represents a distinct operation of the group. For if [Sigma] = [Sigma]1, or S_a^[alpha] S_b^[beta] ... S_d^[delta] = S_(a1)^([alpha]1) S_(b1)^([beta]1) ... S_(d1)^([delta]1), then S_(d1)^(-[delta]1) ... S_(b1)^(-[beta]1) S_(a1)^(-[alpha]1) S_a^[alpha] S_b^[beta] ... S_d^[delta] = 1; and unless a = a1, b = b1, ..., [alpha] = [alpha]1, [beta] = [beta]1, ..., this is a relation connecting the generating operations.

Suppose now that T1, T2, ... are operations of G, and that H is that self-conjugate subgroup of G which is generated by T1, T2, ... and the operations conjugate to them. Then, of the operations that can be formed from S1, S2, ..., S_n, the set [Sigma]H, and no others, reduce to the same operation [Sigma] when the conditions T1 = 1, T2 = 1, ... are satisfied by the generating operations. Hence the group which is generated by the given operations, when subjected to the conditions just written, is simply isomorphic with the factor-group G/H. Moreover, this is obviously true even when the conditions are such that the generating operations are no longer independent. Hence any discontinuous group may be defined abstractly, that is, in regard to the laws of combination of its operations apart from their actual form, by a set of generating operations and a system of relations connecting them. Conversely, when such a set of operations and system of relations are given arbitrarily they define in abstract form a single discontinuous group. It may, of course, happen that the group so defined is a group of finite order, or that it reduces to the identical operation only; but in regard to the general statement these will be particular and exceptional cases.

Properly and improperly discontinuous groups.

An operation of a discontinuous group must necessarily be specified analytically by a system of equations of the form

x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n),

and the different operations of the group will be given by different sets of values of the parameters a1, a2, ..., a_r. No one of these parameters is susceptible of continuous variations, but at least one must be capable of taking a number of values which is not finite, if the group is not one of finite order. Among the sets of values of the parameters there must be one which gives the identical transformation. No other transformation makes each of the differences x'1 - x1, x'2 - x2, ..., x'_n - x_n vanish. Let d be an arbitrary assigned positive quantity. Then if a transformation of the group can be found such that the modulus of each of these differences is less than d when the variables have arbitrary values within an assigned range of variation, however small d may be chosen, the group is said to be _improperly_ discontinuous. In the contrary case the group is called _properly_ discontinuous. The range within which the variables are allowed to vary may clearly affect the question whether a given group is properly or improperly discontinuous. For instance, the group defined by the equation x' = ax + b, where a and b are any rational numbers, is improperly discontinuous; and the group defined by x' = x + a, where a is an integer, is properly discontinuous, whatever the range of the variable. On the other hand, the group, to be later considered, defined by the equation x' = (ax + b)/(cx + d), where a, b, c, d are integers satisfying the relation ad - bc = 1, is properly discontinuous when x may take any complex value, and improperly discontinuous when the range of x is limited to real values.

Linear discontinuous groups.

Among the discontinuous groups that occur in analysis, a large number may be regarded as arising by imposing limitations on the range of variation of the parameters of continuous groups. If

x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n),

are the finite equations of a continuous group, and if C with parameters c1, c2, ..., c_r is the operation which results from carrying out A and B with corresponding parameters in succession, then the c's are determined uniquely by the a's and the b's. If the c's are rational functions of the a's and b's, and if the a's and b's are arbitrary rational numbers of a given corpus (see NUMBER), the c's will be rational numbers of the same corpus. If the c's are rational integral functions of the a's and b's, and the latter are arbitrarily chosen integers of a corpus, then the c's are integers of the same corpus. Hence in the first case the above equations, when the a's are limited to be rational numbers of a given corpus, will define a discontinuous group; and in the second case they will define such a group when the a's are further limited to be integers of the corpus. A most important class of discontinuous groups are those that arise in this way from the general linear continuous group in a given set of variables. For n variables the finite equations of this continuous group are

x'_s = a_(s1)x1 + a_(s2)x2 + ... + a_(sn)x_n, (s = 1, 2, ..., n),

where the determinant of the a's must not be zero. In this case the c's are clearly integral lineo-linear functions of the a's and b's. Moreover, the determinant of the c's is the product of the determinant of the a's and the determinant of the b's. Hence equations (ii.), where the parameters are restricted to be integers of a given corpus, define a discontinuous group; and if the determinant of the coefficients is limited to the value unity, they define a discontinuous group which is a (self-conjugate) subgroup of the previous one.

The simplest case which thus presents itself is that in which there are two variables while the coefficients are rational integers. This is the group defined by the equations

x' = ax + by, \ > y' = cx + dy, /

where a, b, c, d are integers such that ad - bc = 1. To every operation of this group there corresponds an operation of the set defined by

az + b z' = ------, cz + d

in such a way that to the product of two operations of the group there corresponds the product of the two analogous operations of the set. The operations of the set (iv.), where ad - bc = 1, therefore constitute a group which is isomorphic with the previous group. The isomorphism is multiple, since to a single operation of the second set there correspond the two operations of the first for which a, b, c, d and -a, -b, -c, -d are parameters. These two groups, which are of fundamental importance in the theory of quadratic forms and in the theory of modular functions, have been the object of very many investigations.

Discontinuous groups arising from geometrical operations.

Another large class of discontinuous groups, which have far-reaching applications in analysis, are those which arise in the first instance from purely geometrical considerations. By the combination and repetition of a finite number of geometrical operations such as displacements, projective transformations, inversions, &c., a discontinuous group of such operations will arise. Such a group, as regards the points of the plane (or of space), will in general be improperly discontinuous; but when the generating operations are suitably chosen, the group may be properly discontinuous. In the latter case the group may be represented in a graphical form by the division of the plane (or space) into regions such that no point of one region can be transformed into another point of the same region by any operation of the group, while any given region can be transformed into any other by a suitable transformation. Thus, let ABC be a triangle bounded by three circular arcs BC, CA, AB; and consider the figure produced from ABC by inversions in the three circles of which BC, CA, AB are part. By inversion at BC, ABC becomes an equiangular triangle A'BC. An inversion in AB changes ABC and A'BC into equiangular triangles ABC' and A"BC'. Successive inversions at AB and BC then will change ABC into a series of equiangular triangles with B for a common vertex. These will not overlap and will just fill in the space round B if the angle ABC is a submultiple of two right angles. If then the angles of ABC are submultiples of two right angles (or zero), the triangles formed by any number of inversions will never overlap, and to each operation consisting of a definite series of inversions at BC, CA and AB will correspond a distinct triangle into which ABC is changed by the operation. The network of triangles so formed gives a graphical representation of the group that arises from the three inversions in BC, CA, AB. The triangles may be divided into two sets, those, namely, like A"BC', which are derived from ABC by an even number of inversions, and those like A'BC or ABC' produced by an odd number. Each set are interchanged among themselves by any even number of inversions. Hence the operations consisting of an even number of inversions form a group by themselves. For this group the quadrilateral formed by ABC and A'BC constitutes a region, which is changed by every operation of the group into a distinct region (formed of two adjacent triangles), and these regions clearly do not overlap. Their distribution presents in a graphical form the group that arises by pairs of inversions at BC, CA, AB; and this group is generated by the operation which consists of successive inversions at AB, BC and that which consists of successive inversions at BC, CA. The group defined thus geometrically may be presented in many analytical forms. If x, y and x', y' are the rectangular co-ordinates of two points which are inverse to each other with respect to a given circle, x' and y' are rational functions of x and y, and conversely. Thus the group may be presented in a form in which each operation gives a birational transformation of two variables. If x + iy = z, x' + iy' = z', and if x', y' is the point to which x, y is transformed by any even number of inversions, then z' and z are connected by a linear relation z' = ([alpha]z + [beta])/([gamma]z + [delta]), where [alpha], [beta], [gamma], [delta] are constants (in general complex) depending on the circles at which the inversions are taken. Hence the group may be presented in the form of a group of linear transformations of a single variable generated by the two linear transformations z' = ([alpha]1z + [beta]1)/([gamma]1z + [delta]1), z' = ([alpha]2z + [beta]2)/([gamma]2z + [delta]2), which correspond to pairs of inversions at AB, BC and BC, CA respectively. In particular, if the sides of the triangle are taken to be x = 0, x^2 + y^2 -1 = 0, x^2 + y^2 + 2x = 0, the generating operations are found to be z' = z + 1, z' = -z^(-1); and the group is that consisting of all transformations of the form z' = (az + b)/(cz + d), where ad - bc = 1, a, b, c, d being integers. This is the group already mentioned which underlies the theory of the elliptic modular functions; a modular function being a function of z which is invariant for some subgroup of finite index of the group in question.

The triangle ABC from which the above geometrical construction started may be replaced by a polygon whose sides are circles. If each angle is a submultiple of two right angles or zero, the construction is still effective to give a set of non-overlapping regions, which represent graphically the group which arises from pairs of inversions in the sides of the polygon. In their analytical form, as groups of linear transformations of a single variable, the groups are those on which the theory of automorphic functions depends. A similar construction in space, the polygons bounded by circular arcs being replaced by polyhedra bounded by spherical faces, has been used by F. Klein and Fricke to give a geometrical representation for groups which are improperly discontinuous when represented as groups of the plane.

Group of a linear differential equation.

The special classes of discontinuous groups that have been dealt with in the previous paragraphs arise directly from geometrical considerations. As a final example we shall refer briefly to a class of groups whose origin is essentially analytical. Let

d^_ny d^(n-1)y dy ----- + P1 -------- + ... + P_(n-1) -- + P_ny = 0 dx^_n dx^(n-1) dx

be a linear differential equation, the coefficients in which are rational functions of x, and let y1, y2, ..., y_n be a linearly independent set of integrals of the equation. In the neighbourhood of a finite value x0 of x, which is not a singularity of any of the coefficients in the equation, these integrals are ordinary power-series in x - x0. If the analytical continuations of y1, y2, ..., y_n be formed for any closed path starting from and returning to x0, the final values arrived at when x0 is again reached will be another set of linearly independent integrals. When the closed path contains no singular point of the coefficients of the differential equation, the new set of integrals is identical with the original set. If, however, the closed path encloses one or more singular points, this will not in general be the case. Let y'1, y'2, ..., y'_n be the new integrals arrived at. Since in the neighbourhood of x0 every integral can be represented linearly in terms of y1, y2, ..., y_n, there must be a system of equations

y'1 = a11y1 + a12y2 + ... + a_(1n)y_n,

y'2 = a21y1 + a22y2 + ... + a_(2n)y_n,

. . . . .

y'_n = a_(n1)y1 + a_(n2)y2 + ... + a_(nn)y_n,

where the a's are constants, expressing the new integrals in terms of the original ones. To each closed path described by x0 there therefore corresponds a definite linear substitution performed on the y's. Further, if S1 and S2 are the substitutions that correspond to two closed paths L1 and L2, then to any closed path which can be continuously deformed, without crossing a singular point, into L1 followed by L2, there corresponds the substitution S1S2. Let L1, L2, ..., L_r be arbitrarily chosen closed paths starting from and returning to the same point, and each of them enclosing a single one of the (r) finite singular points of the equation. Every closed path in the plane can be formed by combinations of these r paths taken either in the positive or in the negative direction. Also a closed path which does not cut itself, and encloses all the r singular points within it, is equivalent to a path enclosing the point at infinity and no finite singular point. If S1, S2, S3, ..., S_r are the linear substitutions that correspond to these r paths, then the substitution corresponding to every possible path can be obtained by combination and repetition of these r substitutions, and they therefore generate a discontinuous group each of whose operations corresponds to a definite closed path. The group thus arrived at is called the group of the equation. For a given equation it is unique in type. In fact, the only effect of starting from another set of independent integrals is to transform every operation of the group by an arbitrary substitution, while choosing a different set of paths is equivalent to taking a new set of generating operations. The great importance of the group of the equation in connexion with the nature of its integrals cannot here be dealt with, but it may be pointed out that if all the integrals of the equation are algebraic functions, the group must be a group of finite order, since the set of quantities y1, y2 ..., y_n can then only take a finite number of distinct values.

_Groups of Finite Order._

We shall now pass on to groups of finite order. It is clear that here we must have to do with many properties which have no direct analogues in the theory of continuous groups or in that of discontinuous groups in general; those properties, namely, which depend on the fact that the number of distinct operations in the group is finite.

Let S1, S2, S3, ..., S_N denote the operations of a group G of finite order N, S1 being the identical operation. The tableau

S1, S2, S3, ..., S_N, S1S2, S2S2, S3S3, ..., S_NS2, S1S3, S2S3, S3S3, ..., S_NS3, . . . . . S1S_N, S2S_N, S3S_N, ..., S_NS_N,

when in it each compound symbol S_pS_q is replaced by the single symbol S_r that is equivalent to it, is called the multiplication table of the group. It indicates directly the result of multiplying together in an assigned sequence any number of operations of the group. In each line (and in each column) of the tableau every operation of the group occurs just once. If the letters in the tableau are regarded as mere symbols, the operation of replacing each symbol in the first line by the symbol which stands under it in the pth line is a permutation performed on the set of N symbols. Thus to the N lines of the tableau there corresponds a set of N permutations performed on the N symbols, which includes the identical permutation that leaves each unchanged. Moreover, if S_pS_q = S_r, then the result of carrying out in succession the permutations which correspond to the pth and qth lines gives the permutation which corresponds to the rth line. Hence the set of permutations constitutes a group which is simply isomorphic with the given group.

Every group of finite order N can therefore be represented in concrete form as a transitive group of permutations on N symbols.

Properties of a group which depend on the order.

The order of any subgroup or operation of G is necessarily finite. If T1(= S1), T2, ..., T_n are the operations of a subgroup H of G, and if [Sigma] is any operation of G which is not contained in H, the set of operations [Sigma]T1, [Sigma]T2, ..., [Sigma]T_n, or [Sigma]H, are all distinct from each other and from the operations of H. If the sets H and [Sigma]H do not exhaust the operations of G, and if [Sigma]' is an operation not belonging to them, then the operations of the set [Sigma]'H are distinct from each other and from those of H and [Sigma]H. This process may be continued till the operations of G are exhausted. The order n of H must therefore be a factor of the order N of G. The ratio N/n is called the index of the subgroup H. By taking for H the cyclical subgroup generated by any operation S of G, it follows that the order of S must be a factor of the order of G.

Every operation S is permutable with its own powers. Hence there must be some subgroup H of G of greatest possible order, such that every operation of H is permutable with S. Every operation of H transforms S into itself, and every operation of the set H[Sigma] transforms S into the same operation. Hence, when S is transformed by every operation of G, just N/n distinct operations arise if n is the order of H. These operations, and no others, are conjugate to S within G; they are said to form a set of conjugate operations. The number of operations in every conjugate set is therefore a factor of the order of G. In the same way it may be shown that the number of subgroups which are conjugate to a given subgroup is a factor of the order of G. An operation which is permutable with every operation of the group is called a _self-conjugate_ operation. The totality of the self-conjugate operations of a group forms a self-conjugate Abelian subgroup, each of whose operations is permutable with every operation of the group.

Sylow's theorem.