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 operationsGenerating operations.may be either finite or infinite, but the former case alone will be here considered. Suppose then that S1, S2, ..., Snis a set of independent operations from which a group G is generated. The general operation of the group will be represented by the symbol SαaSβb... Sδd, or Σ, where a, b, ..., d are chosen from 1, 2, ..., n, and α, β, ..., δ are any positive or negative integers. It may be assumed that no two successive suffixes in Σ are the same, for if b = a, then SαaSβbmay be replaced by Sα+βa. If there are no relations connecting the generating operations and the identical operation, every distinct symbol Σ represents a distinct operation of the group. For if Σ = Σ1, or SαaSβb... Sδd= Sα1a1Sβ1b1... Sδ1d1, then S−δ1d1... S−β1b1S−α1a1SαaSβb... Sδd= 1; and unless a = a1, b = b1, ..., α = α1, β = β1, ..., 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, ..., Sn, the set ΣH, and no others, reduce to the same operation Σ 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.An operation of a discontinuous group must necessarily be specifiedProperly and improperly discontinuous groups.analytically by a system of equations of the formx′s= ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),and the different operations of the group will be given by different sets of values of the parameters a1, a2, ..., ar. 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− xnvanish. 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 beimproperlydiscontinuous. In the contrary case the group is calledproperlydiscontinuous. 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 groupdefined 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.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. Ifx′s= ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),are the finite equations of a continuous group, and if C with parameters c1, c2, ..., cris 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 (seeNumber), 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 whenLinear discontinuous groups.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 arex′s= as1x1+ as2x2+ ... + asnxn, (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 equationsx′ = 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 byz′ =az + b,cz + din 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.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 geometricalDiscontinuous groups arising from geometrical operations.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′ = (αz + β) / (γz + δ), where α, β, γ, δ 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′ = (α1z + β1) / (γ1z + δ1), z′ = (α2z + β2) / (γ2z + δ2), 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, x2+ y2− 1 = 0, x2+ y2+ 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.The special classes of discontinuous groups that have been dealt withGroup of a linear differential equation.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. Letdny+ P1dn−1y+ ... + Pn−1dy+ Pny = 0dxndxn−1dxbe a linear differential equation, the coefficients in which are rational functions of x, and let y1, y2, ..., ynbe a linearly independent set of integrals of the equation. In the neighbourhood of a finite value x0of 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, ..., ynbe formed for any closed path starting from and returning to x0, the final values arrived at when x0is 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′nbe the new integrals arrived at. Since in the neighbourhood of x0every integral can be represented linearly in terms of y1, y2, ..., yn, there must be a system of equationsy′1= a11y1+ a12y2+ ... + a1nyn,y′2= a21y1+ a22y2+ ... + a2nyn,· · · ·y′n= an1y1+ an2y2+ ... + annyn,where the a’s are constants, expressing the new integrals in terms of the original ones. To each closed path described by x0there therefore corresponds a definite linear substitution performed on the y’s. Further, if S1and S2are the substitutions that correspond to two closed paths L1and L2, then to any closed path which can be continuously deformed, without crossing a singular point, into L1followed by L2, there corresponds the substitution S1S2. Let L1, L2, ..., Lrbe 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, ..., Srare 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..., yncan then only take a finite number of distinct values.
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 operationsGenerating operations.may be either finite or infinite, but the former case alone will be here considered. Suppose then that S1, S2, ..., Snis a set of independent operations from which a group G is generated. The general operation of the group will be represented by the symbol SαaSβb... Sδd, or Σ, where a, b, ..., d are chosen from 1, 2, ..., n, and α, β, ..., δ are any positive or negative integers. It may be assumed that no two successive suffixes in Σ are the same, for if b = a, then SαaSβbmay be replaced by Sα+βa. If there are no relations connecting the generating operations and the identical operation, every distinct symbol Σ represents a distinct operation of the group. For if Σ = Σ1, or SαaSβb... Sδd= Sα1a1Sβ1b1... Sδ1d1, then S−δ1d1... S−β1b1S−α1a1SαaSβb... Sδd= 1; and unless a = a1, b = b1, ..., α = α1, β = β1, ..., 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, ..., Sn, the set ΣH, and no others, reduce to the same operation Σ 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.
An operation of a discontinuous group must necessarily be specifiedProperly and improperly discontinuous groups.analytically by a system of equations of the form
x′s= ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),
and the different operations of the group will be given by different sets of values of the parameters a1, a2, ..., ar. 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− xnvanish. 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 beimproperlydiscontinuous. In the contrary case the group is calledproperlydiscontinuous. 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 groupdefined 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.
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= ƒs(x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),
are the finite equations of a continuous group, and if C with parameters c1, c2, ..., cris 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 (seeNumber), 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 whenLinear discontinuous groups.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= as1x1+ as2x2+ ... + asnxn, (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
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.
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 geometricalDiscontinuous groups arising from geometrical operations.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′ = (αz + β) / (γz + δ), where α, β, γ, δ 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′ = (α1z + β1) / (γ1z + δ1), z′ = (α2z + β2) / (γ2z + δ2), 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, x2+ y2− 1 = 0, x2+ y2+ 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.
The special classes of discontinuous groups that have been dealt withGroup of a linear differential equation.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
be a linear differential equation, the coefficients in which are rational functions of x, and let y1, y2, ..., ynbe a linearly independent set of integrals of the equation. In the neighbourhood of a finite value x0of 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, ..., ynbe formed for any closed path starting from and returning to x0, the final values arrived at when x0is 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′nbe the new integrals arrived at. Since in the neighbourhood of x0every integral can be represented linearly in terms of y1, y2, ..., yn, there must be a system of equations
y′1= a11y1+ a12y2+ ... + a1nyn,y′2= a21y1+ a22y2+ ... + a2nyn,· · · ·y′n= an1y1+ an2y2+ ... + annyn,
y′1= a11y1+ a12y2+ ... + a1nyn,
y′2= a21y1+ a22y2+ ... + a2nyn,
· · · ·
y′n= an1y1+ an2y2+ ... + annyn,
where the a’s are constants, expressing the new integrals in terms of the original ones. To each closed path described by x0there therefore corresponds a definite linear substitution performed on the y’s. Further, if S1and S2are the substitutions that correspond to two closed paths L1and L2, then to any closed path which can be continuously deformed, without crossing a singular point, into L1followed by L2, there corresponds the substitution S1S2. Let L1, L2, ..., Lrbe 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, ..., Srare 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..., yncan 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, ..., SNdenote the operations of a group G of finite order N, S1being the identical operation. The tableauS1,S2,S3,...,SN,S1S2,S2S2,S3S3,...,SNS2,S1S3,S2S3,S3S3,...,SNS3,·····S1SN,S2SN,S3SN,...,SNSN,when in it each compound symbol SpSqis replaced by the single symbol Srthat 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 SpSq= Sr, 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.The order of any subgroup or operation of G is necessarily finite. If T1(= S1), T2, ..., Tnare the operations of a subgroup H of G, and if Σ is any operation of G which is not contained in H,Properties of a group which depend on the order.the set of operations ΣT1, ΣT2, ..., ΣTn, or ΣH, are all distinct from each other and from the operations of H. If the sets H and ΣH do not exhaust the operations of G, and if Σ′ is an operation not belonging to them, then the operations of the set Σ′H are distinct from each other and from those of H and ΣH. 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 HΣ 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 aself-conjugateoperation. 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.An Abelian group contains subgroups whose orders are any given factors of the order of the group. In fact, since every subgroup H of an Abelian group G and the corresponding factor groups G/H areSylow’s theorem.Abelian, this result follows immediately by an induction from the case in which the order contains n prime factors to that in which it contains n + 1. For a group which is not Abelian no general law can be stated as to the existence or non-existence of a subgroup whose order is an arbitrarily assigned factor of the order of the group. In this connexion the most important general result, which is independent of any supposition as to the order of the group, is known as Sylow’s theorem, which states that if pais the highest power of a prime p which divides the order of a group G, then G contains a single conjugate set of subgroups of order pa, the number in the set being of the form 1 + kp. Sylow’s theorem may be extended to show that if pa′is a factor of the order of a group, the number of subgroups of order pa′is of the form 1 + kp. If, however, pa′is not the highest power of p which divides the order, these groups do not in general form a single conjugate set.The importance of Sylow’s theorem in discussing the structure of a group of given order need hardly be insisted on. Thus, as a very simple instance, a group whose order is the product p1p2of two primes (p1< p2) must have a self-conjugate subgroup of order p2, since the order of the group contains no factor, other than unity, of the form 1 + kp2. The same again is true for a group of order p12p2, unless p1= 2, and p2= 3.There is one other numerical property of a group connected with its order which is quite general. If N is the order of G, and n a factor of N, the number of operations of G, whose orders are equal to or are factors of n, is a multiple of n.As already defined, a composite group is a group which contains one or more self-conjugate subgroups, whose orders are greater than unity. If H is a self-conjugate subgroup of G, the factor-groupComposition-series of a group.G/H may be either simple or composite. In the former case G can contain no self-conjugate subgroup K, which itself contains H; for if it did K/H would be a self-conjugate subgroup of G/H. When G/H is simple, H is said to be a maximum self-conjugate subgroup of G. Suppose now that G being a given composite group, G, G1, G2, ..., Gn, 1 is a series of subgroups of G, such that each is a maximum self-conjugate subgroup of the preceding; the last term of the series consisting of the identical operation only. Such a series is called acomposition-seriesof G. In general it is not unique, since a group may have two or more maximum self-conjugate subgroups. A composition-series of a group, however it may be chosen, has the property that the number of terms of which it consists is always the same, while the factor-groups G/G1, G1/G2, ..., Gndiffer only in the sequence in which they occur. It should be noticed that though a group defines uniquely the set of factor-groups that occur in its composition-series, the set of factor-groups do not conversely in general define a single type of group. When the orders of all the factor-groups are primes the group is said to besoluble.If the series of subgroups G, H, K, ..., L, 1 is chosen so that each is the greatest self-conjugate subgroup of G contained in the previous one, the series is called a chief composition-series of G. All such series derived from a given group may be shown to consist of the same number of terms, and to give rise to the same set of factor-groups, except as regards sequence. The factor-groups of such a series will not, however, necessarily be simple groups. From any chief composition-series a composition-series may be formed by interpolating between any two terms H and K of the series for which H/K is not a simple group, a number of terms h1, h2, ..., hr; and it may be shown that the factor-groups H/h1, h1/h2, ..., hr/K are all simply isomorphic with each other.A group may be represented as isomorphic with itself by transforming all its operations by any one of them. In fact, if SpSq= Sr, then S−1SpS·S−1SqS = S−1SrS. An isomorphism of theIsomorphism of a group with itself.group with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation carried out on the symbols of the operations, being indeed a permutation performed on these symbols. The totality of these operations clearly constitutes a group isomorphic with the given group, and this group is called the group of inner isomorphisms. A group is simply or multiply isomorphic with its group of inner isomorphisms according as it does not or does contain self-conjugate operations other than identity. It may be possible to establish a correspondence between the operations of a group other than those given by the inner isomorphisms, such that if S′ is the operation corresponding to S, then S′pS′q= S′ris a consequence of SpSq= Sr. The substitution on the symbols of the operations of a group resulting from such a correspondence is called an outer isomorphism. The totality of the isomorphisms of both kinds constitutes the group of isomorphisms of the given group, and within this the group of inner isomorphisms is a self-conjugate subgroup. Every set of conjugate operations of a group is necessarily transformed into itself by an inner isomorphism, but two or more sets may be interchanged by an outer isomorphism.A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called acharacteristicsubgroup. A series of groups G, G1, G2, ..., 1, such that each is a maximum characteristic subgroup of G contained in the preceding, may be shown to have the same invariant properties as the subgroups of a composition series. A group which has no characteristic subgroup must be either a simplegroup or the direct product of a number of simply isomorphic simple groups.It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general suchPermutation-groups.a representation is possible with a smaller number of symbols. Let H be a subgroup of G, and let the operations of G be divided, in respect of H, into the sets H, S2H, S3H, ..., SmH. If S is any operation of G, the sets SH, SS2H, SS3H, ..., SSmH differ from the previous sets only in the sequence in which they occur. In fact, if SSpbelong to the set SqH, then since H is a group, the set SSpH is identical with the set SqH. Hence, to each operation S of the group will correspond a permutation performed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. The number of symbols involved in such a representation is called thedegreeof the group. In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same.The total number of permutations that can be performed on n symbols is n!, and these necessarily constitute a group. It is known as thesymmetricgroup of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions. When any permutation is carried out on the product of the n(n − 1)/2, differences of the n symbols, it must either remain unaltered or its sign must be changed. Those permutations which leave the product unaltered constitute a group of order n!/2, which is called thealternatinggroup of degree n; it is a self-conjugate subgroup of the symmetric group. Except when n = 4 the alternating group is a simple group. A group of degree n, which is not contained in the alternating group, must necessarily have a self-conjugate subgroup of index 2, consisting of those of its permutations which belong to the alternating group.Among the various concrete forms in which a group of finite order can be presented the most important is that of a group of linearGroups of linear substitutions.substitutions. Such groups have already been referred to in connexion with discontinuous groups. Here the number of distinct substitutions is necessarily finite; and to each operation S of a group G of finite order there will correspond a linear substitution s, viz.xi=Σj=mj=1sijxj(i, j = 1, 2, ..., m),on a set of m variables, such that if ST = U, then st = u. The linear substitutions s, t, u, ... then constitute a group g with which G is isomorphic; and whether the isomorphism is simple or multiple g is said to give a “representation” of G as a group of linear substitutions. If all the substitutions of g are transformed by the same substitution on the m variables, the (in general) new group of linear substitutions so constituted is said to be “equivalent” with g as a representation of G; and two representations are called “non-equivalent,” or “distinct,” when one is not capable of being transformed into the other.A group of linear substitutions on m variables is said to be “reducible” when it is possible to choose m′ (< m) linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called “irreducible.” It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.It has been seen at the beginning of this section that every group of finite order N can be presented as a group of permutations (i.e.linear substitutions in a limited sense) on N symbols. This group is obviously reducible; in fact, the sum of the symbols remain unaltered by every substitution of the group. The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following.If r is the number of different sets of conjugate operations in the group, then, when the group of N permutations is completely reduced,(i.) just r distinct irreducible representations occur:(ii.) each of these occurs a number of times equal to the number of symbols on which it operates:(iii.) these irreducible representations exhaust all the distinct irreducible representations of the group.Among these representations what is called the “identical” representation necessarily occurs,i.e.that in which each operation of the group corresponds to leaving a single symbol unchanged. If these representations are denoted by Γ1, Γ2, ..., Γr, then any representation of the group as a group of linear substitutions, or in particular as a group of permutations, may be uniquely represented by a symbol ΣαiΓi, in the sense that the representation when completely reduced will contain the representation Γijust αitimes for each suffix i.A representation of a group of finite order as an irreducible groupGroup characteristics.of linear substitutions may be presented in an infinite number of equivalent forms. Ifx′i= Σsijxj(i, j = 1, 2, ..., m),is the linear substitution which, in a given irreducible representation of a group of finite order G, corresponds to the operation S, the determinants11− λs12...s1ms21s22− λ...s2m..................sm1s2m...smm− λis invariant for all equivalent representations, when written as a polynomial in λ. Moreover, it has the same value for S and S′, if these are two conjugate operations in G. Of the various invariants that thus arise the most important is s11+ s22+ ... + smm, which is called the “characteristic” of S. If S is an operation of order p, its characteristic is the sum of m pth roots of unity; and in particular, if S is the identical operation its characteristic is m. If r is the number of sets of conjugate operations in G, there is, for each representation of G as an irreducible group, a set of r characteristics: X1, X2, ... Xr, one corresponding to each conjugate set; so that for the r irreducible representations just r such sets of characteristics arise. These are distinct, in the sense that if Ψ1, Ψ2, ..., Ψrare the characteristics for a distinct representation from the above, then Xiand Ψiare not equal for all values of the suffix i. It may be the case that the r characteristics for a given representation are all real. If this is so the representation is said to be self-inverse. In the contrary case there is always another representation, called the “inverse” representation, for which each characteristic is the conjugate imaginary of the corresponding one in the original representation. The characteristics are subject to certain remarkable relations. If hpdenotes the number of operations in thepth conjugate set, while Xip, and Xjpare the characteristics of thepth conjugate set in Γiand Γj, thenΣp=rp=1hpXipXjp= 0 or n,according to Γiand Γjare not or are inverse representations, n being the order of G.AgainΣi=ri=1XipXiq= 0 or n/hpaccording as the pth and qth conjugate sets are not or are inverse; the qth set being called the inverse of thepth if it consists of the inverses of the operations constituting thepth.Another form in which every group of finite order can be representedLinear homogeneous groups.is that known as a linear homogeneous group. If in the equationsx′r= ar1x1+ ar2x2+ ... + armxm, (r = 1, 2, ..., m),which define a linear homogeneous substitution, the coefficients are integers, and if the equations are replaced by congruences to a finite modulus n, the system of congruences will give a definite operation, provided that the determinant of the coefficients is relatively prime to n. The product of two such operations is another operation of the same kind; and the total number of distinct operations is finite, since there is only a limited number of choices for the coefficients. The totality of these operations, therefore, constitutes a group of finite order; and such a group is known as alinear homogeneousgroup. If n is a prime the order of the group is(nm− 1) (nm− n) ... (nm− nm−1).The totality of the operations of the linear homogeneous group for which the determinant of the coefficients is congruent to unity forms a subgroup. Other subgroups arise by considering those operations which leave a function of the variables unchanged (mod. n). All such subgroups are known as linear homogeneous groups.When the ratios only of the variables are considered, there arises alinear fractionalgroup, with which the corresponding linear homogeneous group is isomorphic. Thus, if p is a prime the totality of the congruencesz′ ≡az + b, ad − bc ≠ 0, (mod. p)cz + dconstitutes a group of order p(p2− 1). This class of groups for various values of p is almost the only one which has been as yet exhaustively analysed. For all values of p except 3 it contains a simple self-conjugate subgroup of index 2.A great extension of the theory of linear homogeneous groups has been made in recent years by considering systems of congruences of the formx′r≡ ar1x1+ ar2x2+ ... + armxm, (r = 1, 2, ..., m),in which the coefficients ars, are integral functions with real integral coefficients of a root of an irreducible congruence to a prime modulus. Such a system of congruences is obviously limited in numbers and defines a group which contains as a subgroup the group defined by the same congruences with ordinary integral coefficients.The chief application of the theory of groups of finite order is to the theory of algebraic equations. The analogy of equations of the second, third and fourth degrees would give rise to theApplications.expectation that a root of an equation of any finite degree could be expressed in terms of the coefficients by a finite number of the operations of addition, subtraction, multiplication, division, and the extraction of roots; in other words, that the equation could be solved by radicals. This, however, as proved by Abel and Galois, is not the case: an equation of a higher degree than the fourth in general defines an algebraic irrationality which cannot be expressed by means of radicals, and the cases in which such an equation can be solved by radicals must be regarded as exceptional. The theory of groups gives the means of determining whether an equation comes under this exceptional case, and of solving the equation when it does. When it does not, the theory provides the means of reducing the problem presented by the equation to a normal form. From this point of view the theory of equations of the fifth degree has been exhaustively treated, and the problems presented by certain equations of the sixth and seventh degrees have actually been reduced to normal form.Galois (seeEquation) showed that, corresponding to every irreducible equation of thenth degree, there exists a transitive substitution-group of degree n, such that every function of the roots, the numerical value of which is unaltered by all the substitutions of the group can be expressed rationally in terms of the coefficients, while conversely every function of the roots which is expressible rationally in terms of the coefficients is unaltered by the substitutions of the group. This group is called the group of the equation. In general, if the equation is given arbitrarily, the group will be the symmetric group. The necessary and sufficient condition that the equation may be soluble by radicals is that its group should be a soluble group. When the coefficients in an equation are rational integers, the determination of its group may be made by a finite number of processes each of which involves only rational arithmetical operations. These processes consist in forming resolvents of the equation corresponding to each distinct type of subgroup of the symmetric group whose degree is that of the equation. Each of the resolvents so formed is then examined to find whether it has rational roots. The group corresponding to any resolvent which has a rational root contains the group of the equation; and the least of the groups so found is the group of the equation. Thus, for an equation of the fifth degree the various transitive subgroups of the symmetric group of degree five have to be considered. These are (i.) the alternating group; (ii.) a soluble group of order 20; (iii.) a group of order 10, self-conjugate in the preceding; (iv.) a cyclical group of order 5, self-conjugate in both the preceding. If x0, x1, x2, x3, x4are the roots of the equation, the corresponding resolvents may be taken to be those which have for roots (i.) the square root of the discriminant; (ii.) the function (x0x1+ x1x2+ x2x3+ x3x4+ x4x0) (x0x2+ x2x4+ x4x1+ x1x3+ x3x0); (iii.) the function x0x1+ x1x2+ x2x3+ x3x4+ x4x0; and (iv.) the function x02x1+ x12x2+ x22x3+ x32x4+ x42x0. Since the groups for which (iii.) and (iv.) are invariant are contained in that for which (ii.) is invariant, and since these are the only soluble groups of the set, the equation will be soluble by radicals only when the function (ii.) can be expressed rationally in terms of the coefficients. If(x0x1+ x1x2+ x2x3+ x3x4+ x4x0) (x0x2+ x2x4+ x4x1+ x1x3+ x3x0)is known, then clearly x0x1+ x1x2+ x2x3+ x3x4+ x4x0can be determined by the solution of a quadratic equation. Moreover, the sum and product (x0+ εx1+ ε2x2+ ε3x3+ ε4x4)5and (x0+ ε4x1+ ε3x2+ ε2x3+ εx4)5can be expressed rationally in terms of x0x1+ x1x2+ x2x3+ x3x4+ x4x0, ε, and the symmetric functions; ε being a fifth root of unity. Hence (x0+ εx1+ ε2x2+ ε3x3+ ε4X4)5can be determined by the solution of a quadratic equation. The roots of the original equation are then finally determined by the extraction of a fifth root. The problem of reducing an equation of the fifth degree, when not soluble by radicals, to a normal form, forms the subject of Klein’sVorlesungen über das Ikosaeder. Another application of groups of finite order is to the theory of linear differential equations whose integrals are algebraic functions. It has been already seen, in the discussion of discontinuous groups in general, that the groups of such equations must be groups of finite order. To every group of finite order which can be represented as an irreducible group of linear substitutions on n variables will correspond a class of irreducible linear differential equations of thenth order whose integrals are algebraic. The complete determination of the class of linear differential equations of the second order with all their integrals algebraic, whose group has the greatest possible order, viz. 120, has been carried out by Klein.Authorities.—Continuous groups:Lie and Engel,Theorie der Transformationsgruppen(Leipzig, vol. i., 1888; vol. ii., 1890; vol. iii., 1893); Lie and Scheffers,Vorlesungen über gewöhnliche Differentialgleichungen mit bekannten infinitesimalen Transformationen(Leipzig, 1891);Idem, Vorlesungen über continuierliche Gruppen(Leipzig, 1893);Idem, Geometrie der Berührungstransformationen(Leipzig, 1896); Klein and Schilling,Höhere Geometrie, vol. ii. (lithographed) (Göttingen, 1893, for both continuous and discontinuous groups). Campbell,Introductory Treatise on Lie’s Theory of Finite Continuous Transformation Groups(Oxford, 1903). Discontinuous groups: Klein and Fricke,Vorlesungen über die Theorie der elliptischen Modulfunktionen(vol. i., Leipzig, 1890) (for a full discussion of the modular group);Idem, Vorlesungen über die Theorie der automorphen Funktionen(vol. i., Leipzig, 1897; vol. ii. pt. i., 1901) (for the general theory of discontinuous groups); Schoenflies,Krystallsysteme und Krystallstruktur(Leipzig, 1891) (for discontinuous groups of motions);Groups of finite order:Galois,Œuvres mathématiques(Paris, 1897, reprint); Jordan,Traité des substitutions et des équations algébriques(Paris, 1870); Netto,Substitutionentheorie und ihre Anwendung auf die Algebra(Leipzig, 1882; Eng. trans. by Cole, Ann Arbor, U.S.A., 1892); Klein,Vorlesungen über das Ikosaeder(Leipzig, 1884; Eng. trans. by Morrice, London, 1888); H. Vogt,Leçons sur la résolution algébrique des équations(Paris, 1895); Weber,Lehrbuch der Algebra(Braunschweig, vol. i., 1895; vol. ii., 1896; a second edition appeared in 1898); Burnside,Theory of Groups of Finite Order(Cambridge, 1897); Bianchi,Teoria dei gruppi di sostituzioni e delle equazioni algebriche(Pisa, 1899); Dickson,Linear Groups with an Exposition of the Galois Field Theory(Leipzig, 1901); De Séguier,Éléments de la théorie des groupes abstraits(Paris, 1904), A summary with many references will be found in theEncyklopädie der mathematischen Wissenschaften(Leipzig, vol. i., 1898, 1899).
Let S1, S2, S3, ..., SNdenote the operations of a group G of finite order N, S1being the identical operation. The tableau
when in it each compound symbol SpSqis replaced by the single symbol Srthat 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 SpSq= Sr, 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.
The order of any subgroup or operation of G is necessarily finite. If T1(= S1), T2, ..., Tnare the operations of a subgroup H of G, and if Σ is any operation of G which is not contained in H,Properties of a group which depend on the order.the set of operations ΣT1, ΣT2, ..., ΣTn, or ΣH, are all distinct from each other and from the operations of H. If the sets H and ΣH do not exhaust the operations of G, and if Σ′ is an operation not belonging to them, then the operations of the set Σ′H are distinct from each other and from those of H and ΣH. 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 HΣ 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 aself-conjugateoperation. 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.
An Abelian group contains subgroups whose orders are any given factors of the order of the group. In fact, since every subgroup H of an Abelian group G and the corresponding factor groups G/H areSylow’s theorem.Abelian, this result follows immediately by an induction from the case in which the order contains n prime factors to that in which it contains n + 1. For a group which is not Abelian no general law can be stated as to the existence or non-existence of a subgroup whose order is an arbitrarily assigned factor of the order of the group. In this connexion the most important general result, which is independent of any supposition as to the order of the group, is known as Sylow’s theorem, which states that if pais the highest power of a prime p which divides the order of a group G, then G contains a single conjugate set of subgroups of order pa, the number in the set being of the form 1 + kp. Sylow’s theorem may be extended to show that if pa′is a factor of the order of a group, the number of subgroups of order pa′is of the form 1 + kp. If, however, pa′is not the highest power of p which divides the order, these groups do not in general form a single conjugate set.
The importance of Sylow’s theorem in discussing the structure of a group of given order need hardly be insisted on. Thus, as a very simple instance, a group whose order is the product p1p2of two primes (p1< p2) must have a self-conjugate subgroup of order p2, since the order of the group contains no factor, other than unity, of the form 1 + kp2. The same again is true for a group of order p12p2, unless p1= 2, and p2= 3.
There is one other numerical property of a group connected with its order which is quite general. If N is the order of G, and n a factor of N, the number of operations of G, whose orders are equal to or are factors of n, is a multiple of n.
As already defined, a composite group is a group which contains one or more self-conjugate subgroups, whose orders are greater than unity. If H is a self-conjugate subgroup of G, the factor-groupComposition-series of a group.G/H may be either simple or composite. In the former case G can contain no self-conjugate subgroup K, which itself contains H; for if it did K/H would be a self-conjugate subgroup of G/H. When G/H is simple, H is said to be a maximum self-conjugate subgroup of G. Suppose now that G being a given composite group, G, G1, G2, ..., Gn, 1 is a series of subgroups of G, such that each is a maximum self-conjugate subgroup of the preceding; the last term of the series consisting of the identical operation only. Such a series is called acomposition-seriesof G. In general it is not unique, since a group may have two or more maximum self-conjugate subgroups. A composition-series of a group, however it may be chosen, has the property that the number of terms of which it consists is always the same, while the factor-groups G/G1, G1/G2, ..., Gndiffer only in the sequence in which they occur. It should be noticed that though a group defines uniquely the set of factor-groups that occur in its composition-series, the set of factor-groups do not conversely in general define a single type of group. When the orders of all the factor-groups are primes the group is said to besoluble.
If the series of subgroups G, H, K, ..., L, 1 is chosen so that each is the greatest self-conjugate subgroup of G contained in the previous one, the series is called a chief composition-series of G. All such series derived from a given group may be shown to consist of the same number of terms, and to give rise to the same set of factor-groups, except as regards sequence. The factor-groups of such a series will not, however, necessarily be simple groups. From any chief composition-series a composition-series may be formed by interpolating between any two terms H and K of the series for which H/K is not a simple group, a number of terms h1, h2, ..., hr; and it may be shown that the factor-groups H/h1, h1/h2, ..., hr/K are all simply isomorphic with each other.
A group may be represented as isomorphic with itself by transforming all its operations by any one of them. In fact, if SpSq= Sr, then S−1SpS·S−1SqS = S−1SrS. An isomorphism of theIsomorphism of a group with itself.group with itself, established in this way, is called an inner isomorphism. It may be regarded as an operation carried out on the symbols of the operations, being indeed a permutation performed on these symbols. The totality of these operations clearly constitutes a group isomorphic with the given group, and this group is called the group of inner isomorphisms. A group is simply or multiply isomorphic with its group of inner isomorphisms according as it does not or does contain self-conjugate operations other than identity. It may be possible to establish a correspondence between the operations of a group other than those given by the inner isomorphisms, such that if S′ is the operation corresponding to S, then S′pS′q= S′ris a consequence of SpSq= Sr. The substitution on the symbols of the operations of a group resulting from such a correspondence is called an outer isomorphism. The totality of the isomorphisms of both kinds constitutes the group of isomorphisms of the given group, and within this the group of inner isomorphisms is a self-conjugate subgroup. Every set of conjugate operations of a group is necessarily transformed into itself by an inner isomorphism, but two or more sets may be interchanged by an outer isomorphism.
A subgroup of a group G, which is transformed into itself by every isomorphism of G, is called acharacteristicsubgroup. A series of groups G, G1, G2, ..., 1, such that each is a maximum characteristic subgroup of G contained in the preceding, may be shown to have the same invariant properties as the subgroups of a composition series. A group which has no characteristic subgroup must be either a simplegroup or the direct product of a number of simply isomorphic simple groups.
It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general suchPermutation-groups.a representation is possible with a smaller number of symbols. Let H be a subgroup of G, and let the operations of G be divided, in respect of H, into the sets H, S2H, S3H, ..., SmH. If S is any operation of G, the sets SH, SS2H, SS3H, ..., SSmH differ from the previous sets only in the sequence in which they occur. In fact, if SSpbelong to the set SqH, then since H is a group, the set SSpH is identical with the set SqH. Hence, to each operation S of the group will correspond a permutation performed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.
The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. The number of symbols involved in such a representation is called thedegreeof the group. In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same.
The total number of permutations that can be performed on n symbols is n!, and these necessarily constitute a group. It is known as thesymmetricgroup of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions. When any permutation is carried out on the product of the n(n − 1)/2, differences of the n symbols, it must either remain unaltered or its sign must be changed. Those permutations which leave the product unaltered constitute a group of order n!/2, which is called thealternatinggroup of degree n; it is a self-conjugate subgroup of the symmetric group. Except when n = 4 the alternating group is a simple group. A group of degree n, which is not contained in the alternating group, must necessarily have a self-conjugate subgroup of index 2, consisting of those of its permutations which belong to the alternating group.
Among the various concrete forms in which a group of finite order can be presented the most important is that of a group of linearGroups of linear substitutions.substitutions. Such groups have already been referred to in connexion with discontinuous groups. Here the number of distinct substitutions is necessarily finite; and to each operation S of a group G of finite order there will correspond a linear substitution s, viz.
xi=Σj=mj=1sijxj(i, j = 1, 2, ..., m),
on a set of m variables, such that if ST = U, then st = u. The linear substitutions s, t, u, ... then constitute a group g with which G is isomorphic; and whether the isomorphism is simple or multiple g is said to give a “representation” of G as a group of linear substitutions. If all the substitutions of g are transformed by the same substitution on the m variables, the (in general) new group of linear substitutions so constituted is said to be “equivalent” with g as a representation of G; and two representations are called “non-equivalent,” or “distinct,” when one is not capable of being transformed into the other.
A group of linear substitutions on m variables is said to be “reducible” when it is possible to choose m′ (< m) linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called “irreducible.” It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.
It has been seen at the beginning of this section that every group of finite order N can be presented as a group of permutations (i.e.linear substitutions in a limited sense) on N symbols. This group is obviously reducible; in fact, the sum of the symbols remain unaltered by every substitution of the group. The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following.
If r is the number of different sets of conjugate operations in the group, then, when the group of N permutations is completely reduced,
(i.) just r distinct irreducible representations occur:
(ii.) each of these occurs a number of times equal to the number of symbols on which it operates:
(iii.) these irreducible representations exhaust all the distinct irreducible representations of the group.
Among these representations what is called the “identical” representation necessarily occurs,i.e.that in which each operation of the group corresponds to leaving a single symbol unchanged. If these representations are denoted by Γ1, Γ2, ..., Γr, then any representation of the group as a group of linear substitutions, or in particular as a group of permutations, may be uniquely represented by a symbol ΣαiΓi, in the sense that the representation when completely reduced will contain the representation Γijust αitimes for each suffix i.
A representation of a group of finite order as an irreducible groupGroup characteristics.of linear substitutions may be presented in an infinite number of equivalent forms. If
x′i= Σsijxj(i, j = 1, 2, ..., m),
is the linear substitution which, in a given irreducible representation of a group of finite order G, corresponds to the operation S, the determinant
is invariant for all equivalent representations, when written as a polynomial in λ. Moreover, it has the same value for S and S′, if these are two conjugate operations in G. Of the various invariants that thus arise the most important is s11+ s22+ ... + smm, which is called the “characteristic” of S. If S is an operation of order p, its characteristic is the sum of m pth roots of unity; and in particular, if S is the identical operation its characteristic is m. If r is the number of sets of conjugate operations in G, there is, for each representation of G as an irreducible group, a set of r characteristics: X1, X2, ... Xr, one corresponding to each conjugate set; so that for the r irreducible representations just r such sets of characteristics arise. These are distinct, in the sense that if Ψ1, Ψ2, ..., Ψrare the characteristics for a distinct representation from the above, then Xiand Ψiare not equal for all values of the suffix i. It may be the case that the r characteristics for a given representation are all real. If this is so the representation is said to be self-inverse. In the contrary case there is always another representation, called the “inverse” representation, for which each characteristic is the conjugate imaginary of the corresponding one in the original representation. The characteristics are subject to certain remarkable relations. If hpdenotes the number of operations in thepth conjugate set, while Xip, and Xjpare the characteristics of thepth conjugate set in Γiand Γj, then
Σp=rp=1hpXipXjp= 0 or n,
according to Γiand Γjare not or are inverse representations, n being the order of G.
Again
Σi=ri=1XipXiq= 0 or n/hp
according as the pth and qth conjugate sets are not or are inverse; the qth set being called the inverse of thepth if it consists of the inverses of the operations constituting thepth.
Another form in which every group of finite order can be representedLinear homogeneous groups.is that known as a linear homogeneous group. If in the equations
x′r= ar1x1+ ar2x2+ ... + armxm, (r = 1, 2, ..., m),
which define a linear homogeneous substitution, the coefficients are integers, and if the equations are replaced by congruences to a finite modulus n, the system of congruences will give a definite operation, provided that the determinant of the coefficients is relatively prime to n. The product of two such operations is another operation of the same kind; and the total number of distinct operations is finite, since there is only a limited number of choices for the coefficients. The totality of these operations, therefore, constitutes a group of finite order; and such a group is known as alinear homogeneousgroup. If n is a prime the order of the group is
(nm− 1) (nm− n) ... (nm− nm−1).
The totality of the operations of the linear homogeneous group for which the determinant of the coefficients is congruent to unity forms a subgroup. Other subgroups arise by considering those operations which leave a function of the variables unchanged (mod. n). All such subgroups are known as linear homogeneous groups.
When the ratios only of the variables are considered, there arises alinear fractionalgroup, with which the corresponding linear homogeneous group is isomorphic. Thus, if p is a prime the totality of the congruences
constitutes a group of order p(p2− 1). This class of groups for various values of p is almost the only one which has been as yet exhaustively analysed. For all values of p except 3 it contains a simple self-conjugate subgroup of index 2.
A great extension of the theory of linear homogeneous groups has been made in recent years by considering systems of congruences of the form
x′r≡ ar1x1+ ar2x2+ ... + armxm, (r = 1, 2, ..., m),
in which the coefficients ars, are integral functions with real integral coefficients of a root of an irreducible congruence to a prime modulus. Such a system of congruences is obviously limited in numbers and defines a group which contains as a subgroup the group defined by the same congruences with ordinary integral coefficients.
The chief application of the theory of groups of finite order is to the theory of algebraic equations. The analogy of equations of the second, third and fourth degrees would give rise to theApplications.expectation that a root of an equation of any finite degree could be expressed in terms of the coefficients by a finite number of the operations of addition, subtraction, multiplication, division, and the extraction of roots; in other words, that the equation could be solved by radicals. This, however, as proved by Abel and Galois, is not the case: an equation of a higher degree than the fourth in general defines an algebraic irrationality which cannot be expressed by means of radicals, and the cases in which such an equation can be solved by radicals must be regarded as exceptional. The theory of groups gives the means of determining whether an equation comes under this exceptional case, and of solving the equation when it does. When it does not, the theory provides the means of reducing the problem presented by the equation to a normal form. From this point of view the theory of equations of the fifth degree has been exhaustively treated, and the problems presented by certain equations of the sixth and seventh degrees have actually been reduced to normal form.
Galois (seeEquation) showed that, corresponding to every irreducible equation of thenth degree, there exists a transitive substitution-group of degree n, such that every function of the roots, the numerical value of which is unaltered by all the substitutions of the group can be expressed rationally in terms of the coefficients, while conversely every function of the roots which is expressible rationally in terms of the coefficients is unaltered by the substitutions of the group. This group is called the group of the equation. In general, if the equation is given arbitrarily, the group will be the symmetric group. The necessary and sufficient condition that the equation may be soluble by radicals is that its group should be a soluble group. When the coefficients in an equation are rational integers, the determination of its group may be made by a finite number of processes each of which involves only rational arithmetical operations. These processes consist in forming resolvents of the equation corresponding to each distinct type of subgroup of the symmetric group whose degree is that of the equation. Each of the resolvents so formed is then examined to find whether it has rational roots. The group corresponding to any resolvent which has a rational root contains the group of the equation; and the least of the groups so found is the group of the equation. Thus, for an equation of the fifth degree the various transitive subgroups of the symmetric group of degree five have to be considered. These are (i.) the alternating group; (ii.) a soluble group of order 20; (iii.) a group of order 10, self-conjugate in the preceding; (iv.) a cyclical group of order 5, self-conjugate in both the preceding. If x0, x1, x2, x3, x4are the roots of the equation, the corresponding resolvents may be taken to be those which have for roots (i.) the square root of the discriminant; (ii.) the function (x0x1+ x1x2+ x2x3+ x3x4+ x4x0) (x0x2+ x2x4+ x4x1+ x1x3+ x3x0); (iii.) the function x0x1+ x1x2+ x2x3+ x3x4+ x4x0; and (iv.) the function x02x1+ x12x2+ x22x3+ x32x4+ x42x0. Since the groups for which (iii.) and (iv.) are invariant are contained in that for which (ii.) is invariant, and since these are the only soluble groups of the set, the equation will be soluble by radicals only when the function (ii.) can be expressed rationally in terms of the coefficients. If
(x0x1+ x1x2+ x2x3+ x3x4+ x4x0) (x0x2+ x2x4+ x4x1+ x1x3+ x3x0)
is known, then clearly x0x1+ x1x2+ x2x3+ x3x4+ x4x0can be determined by the solution of a quadratic equation. Moreover, the sum and product (x0+ εx1+ ε2x2+ ε3x3+ ε4x4)5and (x0+ ε4x1+ ε3x2+ ε2x3+ εx4)5can be expressed rationally in terms of x0x1+ x1x2+ x2x3+ x3x4+ x4x0, ε, and the symmetric functions; ε being a fifth root of unity. Hence (x0+ εx1+ ε2x2+ ε3x3+ ε4X4)5can be determined by the solution of a quadratic equation. The roots of the original equation are then finally determined by the extraction of a fifth root. The problem of reducing an equation of the fifth degree, when not soluble by radicals, to a normal form, forms the subject of Klein’sVorlesungen über das Ikosaeder. Another application of groups of finite order is to the theory of linear differential equations whose integrals are algebraic functions. It has been already seen, in the discussion of discontinuous groups in general, that the groups of such equations must be groups of finite order. To every group of finite order which can be represented as an irreducible group of linear substitutions on n variables will correspond a class of irreducible linear differential equations of thenth order whose integrals are algebraic. The complete determination of the class of linear differential equations of the second order with all their integrals algebraic, whose group has the greatest possible order, viz. 120, has been carried out by Klein.
Authorities.—Continuous groups:Lie and Engel,Theorie der Transformationsgruppen(Leipzig, vol. i., 1888; vol. ii., 1890; vol. iii., 1893); Lie and Scheffers,Vorlesungen über gewöhnliche Differentialgleichungen mit bekannten infinitesimalen Transformationen(Leipzig, 1891);Idem, Vorlesungen über continuierliche Gruppen(Leipzig, 1893);Idem, Geometrie der Berührungstransformationen(Leipzig, 1896); Klein and Schilling,Höhere Geometrie, vol. ii. (lithographed) (Göttingen, 1893, for both continuous and discontinuous groups). Campbell,Introductory Treatise on Lie’s Theory of Finite Continuous Transformation Groups(Oxford, 1903). Discontinuous groups: Klein and Fricke,Vorlesungen über die Theorie der elliptischen Modulfunktionen(vol. i., Leipzig, 1890) (for a full discussion of the modular group);Idem, Vorlesungen über die Theorie der automorphen Funktionen(vol. i., Leipzig, 1897; vol. ii. pt. i., 1901) (for the general theory of discontinuous groups); Schoenflies,Krystallsysteme und Krystallstruktur(Leipzig, 1891) (for discontinuous groups of motions);Groups of finite order:Galois,Œuvres mathématiques(Paris, 1897, reprint); Jordan,Traité des substitutions et des équations algébriques(Paris, 1870); Netto,Substitutionentheorie und ihre Anwendung auf die Algebra(Leipzig, 1882; Eng. trans. by Cole, Ann Arbor, U.S.A., 1892); Klein,Vorlesungen über das Ikosaeder(Leipzig, 1884; Eng. trans. by Morrice, London, 1888); H. Vogt,Leçons sur la résolution algébrique des équations(Paris, 1895); Weber,Lehrbuch der Algebra(Braunschweig, vol. i., 1895; vol. ii., 1896; a second edition appeared in 1898); Burnside,Theory of Groups of Finite Order(Cambridge, 1897); Bianchi,Teoria dei gruppi di sostituzioni e delle equazioni algebriche(Pisa, 1899); Dickson,Linear Groups with an Exposition of the Galois Field Theory(Leipzig, 1901); De Séguier,Éléments de la théorie des groupes abstraits(Paris, 1904), A summary with many references will be found in theEncyklopädie der mathematischen Wissenschaften(Leipzig, vol. i., 1898, 1899).