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GROUPS, THEORY

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Originally appearing in Volume V12, Page 634 of the 1911 Encyclopedia Britannica.
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GROUPS, THEORY OF be represented by Xi, (i=--I, 2, ... , r), then the increment of F is given by (e1Xi+e2X,+ ... +e,X,)Fbt. When the equations (i.) defining the See also:general operation of the See also:group are given, the coefficients cafe/aai which enter in these See also:differential operators are functions of the variables which can be directly calculated. The differential operator eiXi+e2X2+ ... +e,X, may then be regarded as defining the most general infinitesimal operation of the group. In fact, if it be for a moment represented by X, then (i+btX)F is the result of carrying out the infinitesimal operation on F; and by putting x2, ... , x„ in turn for F, the actual infinitesimal operation is reproduced. By a very convenient, though perhaps hardly justifiable, phraseology this differential operator is itself spoken of as the general infinitesimal operation of the group. The sense in which this phraseology is to be understood will be made clear by the foregoing explanations. We suppose now that the constants ei, e2, ... , e, have assigned values.

Then the result of repeating the particular infinitesimal operation eiXi+e2X2+ . . . +e,X, or X an See also:

infinite number of times is some finite operation of the group. The effect of this finite operation on F may be directly calculated. In fact, if bt is the infinitesimal already introduced, then z d = X.F, dt, = X.X.F,. F'=F+tdF+ tz d2F+ .. dt 1.2 dt2 =F+tX.F+I22X.X.F+ .. . It must, of course, be understood that in this See also:analytical See also:representation of the effect of the finite operation on F it is implied that l is taken sufficiently small to ensure the convergence of the (in general) infinite See also:series. When xi, x2, . . . are written in turn for F, the See also:system of equations x'a=(I+tX+ 2X.X+... )x,, (s=I,2, ... , n) (ii.) represent the finite operation completely. If t is here regarded as a parameter, this set of operations must in themselves constitute a group, since they arise by the repetition of a single infinitesimal operation.

That this is really the See also:

case results immediately from noticing that the result of eliminating F' between F'=F+tX.F+ - X.X.F+ .. . ~x and F",= F'+t'X.F'+i—2X.X.F'+ .. is Fr=F+(t+t')X.F+(t 12')zX.X.F+ .. The group thus generated by the repetition of an infinitesimal operation is called a cyclical group; so that a continuous group contains a cyclical subgroup corresponding to each of its infinitesimal operations. The system of equations (ii.) represents an operation of the group whatever the constants ei, e2, . , e, may be. Hence if eit, ezt, .. , ert be replaced by al, See also:a2, ... , a,. the equations (ii.) represent a set of operations, depending on r parameters and belonging to the group. They must therefore be a See also:form of the general equations for any operation of the group, and are See also:equivalent to the equations (i.). The determination of the finite equations of a cyclical group, when the infinitesimal operation which generates it is given, will always depend on the integration of a set of simultaneous See also:ordinary differential equations. As a very See also:simple example we may consider the case in which the infinitesimal ooeration is given by X =x2a/ax, so that there is only a single variable.

The relation between x' and t is given by dx'/dt =x'2, with the See also:

condition that x'=x when t=0. This gives at once x' =x/(I-tx), which might also be obtained by the See also:direct use of (ii.). When the finite equations (i.) of a continuous group of See also:order r are known, it has now been seen that the differential operator which defines the most general infinitesimal operation of the Relations group can be directly constructed, and that it contains r between arbitrary constants. This is equivalent to saying that theca- the group contains r linearly See also:independent infinitesimal finitesimal operations; and that the most general infinitesimal operations operation is obtained by combining these linearly with of a finite See also:constant coefficients. Moreover, when any r independent continuous infinitesimal operations of the group are known, it has group. been seen how the general finite operation of the group may be calculated. This obviously suggests that it must be possible to define the group by means of its infinitesimal operations alone; and it is clear that such a See also:definition would lend itself more readily to some applications (for instance, to the theory of differential equations) than the definition by means of the finite equations. On the other See also:hand, r arbitrarily given linear differential operators will not, in general, give rise to a finite continuous group of order r; and the question arises as to what conditions such a set of operators Hence must satisfy in order that they may, in fact, be the independent infinitesimal operations of such a group. If X, Y are two linear differential operators, XY - YX is also a linear differential operator. It is called the " combinant " of X and Y (See also:Lie uses the expression Klammerausdruck) and is denoted by (XY). If X, Y, Z are any three linear differential operators the identity (known as See also:Jacobi's) (X (YZ)) + (Y (ZX)) + (Z (XY)) =0 holds between them. Now it may be shown that any continuous group of which X, Y are infinitesimal operations contains also (XY) among its infinitesimal operations. Hence if r linearly independent operations Xi, X2, ...

, X, give rise to a finite continuous group of order r, the combinant of each pair must be expressible linearly in terms of the r operations themselves : that is, there must be a system of relations (XIXi) = CiikXk, k—i where the c's are constants. Moreover, from Jacobi's identity and the identity (XY)+(YX) =o it follows that the c's are subject to the relations Ciie+Gilt =0, and Z(cjk8Ciat+Cki,Cjst+CijaCkat) =0 (iii.) for all values of i, j, k and t. The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in order Determinathat Xi, X2, . . . , Xr may generate, as infinitesimal See also:

lion of the operations, a continuous group of order r, are also lion of sufficient, distinct of For the See also:proof of this fundamental theorem see Lie's types continuous See also:works (cf. Lie-See also:Engel, i. See also:chap. 9; iii. chap. 25). groups of If two continuous groups of order r are such that, for a given each, a set of linearly independent infinitesimal operations order. Xi, X2, . . , X, and Yi, Y2, ... , Yr can be chosen, so that in the relations (X,Xj) =~ci;See also:aXe, (YiYi) =~diiaya, the constants ciia and diia are the same for all values of i, j and s, the two groups are simply isomorphic, X. and Y. being corresponding infinitesimal operations. Two continuous groups of order r, whose infinitesimal operations obey the same system of equations (iii.), may be of very different form; for instance, the number of variables for the one may be different from that for the other.

They are, however, said to be of the same type, in the sense that the See also:

laws according to which their operations combine are the same for both. The problem..of determining all distinct types of groups of order r is then contained in the purely algebraical problem of finding all the systems of r3 quantities ciia which satisfy the relations Cite+Ciie = 0, (cijecekt +Cjs,Cait +Ckiacejt) =0, for all values of i,,7, It and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, Xi, X2, . . , X, may be replaced by any r independent linear functions of themselves, and the c's will then be transformed by a 'linear substitution containing See also:r2 independent parameters. This, however, does not alter the type of group considered. For a single parameter there is, of course, only one type of group, which has been called cyclical. For a group of order two there is a single relation (X1X2) = aXi+133X2. If a and 0 are not both zero, let a be finite. The relation may then be written (aXi+/3X2, a-1)(2) = aXi$X2. Hence if aXi+QX2X'i, and a1X2=X'2, then (X'iX'2) =Xi'. There are, therefore, just two types of group of order two, the one given by the relation last written, and the other by (X1X2) =0.

Lie has determined all distinct types of continuous groups of orders three or four; and all types of non-integrable groups (a See also:

term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744). A problem of fundamental importance in connexion with any given continuous group is the determination of the self-conjugate Self-consubgroups which it contains. If X is an infinitesimal Fugate operation of a group, and Y any other, the general form subgroups. of the infinitesimal operations which are conjugate to X is integrable X+t(XY)+I 2((XY)Y)+ ... groups. Any subgroup which contains all the operations conjugate to X must therefore contain all infinitesimal operations (X13), ((XY)Y) , .. where for Y each infinitesimal operation of the group is taken in turn. Hence if X'i, X'2, . . . , X', are s linearly independent operations of the group which generate a self-conjugate subgroup of order s, then for every infinitesimal operation Y of the group relations of the form ama (X'iY) = E a1,X'e, (i =I, 2, - .. , s) 0.4 must be satisfied. Conversely, if such a set of relations is satisfied, X'1, X'2, ... , X', generate a subgroup of order s, which contains every operation conjugate to each of the infinitesimal generating operations, and is therefore a self-conjugate subgroup.

A specially important self-conjugate subgroup is that generated by the combinants of the r infinitesimal generating operations. That these generate a self-conjugate subgroup follows from the relations (iii.). In fact, ((X1Xi)Xk) =ECiis(XsXk). s Of the ar(r-I) combinants not more than r can be linearly independent. When exactly r of them are linearly independent, the self-conjugate group generated by them coincides with the See also:

original group. If the number that are linearly independent is less than r, the self-conjugate subgroup generated by them is actually a subgroup; i.e. its order is less than that of the original group. This subgroup is known as the derived group, and Lie has called a group perfect when it coincides with its derived group. A simple group, since it contains no self-conjugate subgroup distinct from itself, is necessarily a perfect group. If G is a given continuous group, GI the derived group of G, G2 that of GI, and so on, the series of groups G, GI, G2, . . will terminate either with the identical operation or with a perfect group; for the order of Gs+1 is less than that of G., unless Ga is a perfect group. When the series terminates with the identical operation, G is said to be an integrable group; in the contrary case G is called non- integrable. If G is an integrable group of order r, the infinitesimal operations XI, X2, .... X, which generate the group may be chosen so that XI, X2, ..., X,1, (r1<r) generate the first derived group, XI, X2, ..., (r2<rl) the second derived group, and so on.

When they are so chosen the constants c;is are clearly such that if rp..1, r2<j~ rq+1, ps.q, then c;is vanishes unless s~ r9+1• In particular the generating operations may be chosen so that cii, vanishes unless s is equal to or less than the smaller of the two See also:

numbers i, j; and conversely, if the c's satisfy these relations, the group is integrable. A simple group, as already defined, is one which has no self-conjugate subgroup. It is a remarkable fact that the determination simple of all distinct types of simple continuous groups has been made, for in the case of discontinuous groups and groups grOUPs. of finite order this is far from being the case. Lie has demonstrated the existence of four See also:great classes of simple groups: (i.) The groups simply isomorphic with the general projective group in space of n dimensions. Such a group is defined analytically as the totality of the transformations of the form as, 1x1+ae, 2x2+... +as, „x„+a„ +1 , ( ) x,,= an+1, 1x1+an+i, 2x2-i- +an+1, nxn+I s=l, 2,.. n , where the a's are parameters. The order of this group is clearly n(n+2). (ii.) The groups simply isomorphic with the totality of the projective transformations which transform a non-See also:special linear complex in space of 2n--I dimensions with itself. The order of this group is n(2n+I). (iii.) and (iv.) The groups simply isomorphic with the totality of the projective transformations which See also:change a See also:quadric of non-vanishing discriminant into itself. These fall into two distinct classes of types according as n is even or See also:odd. In either case the order is Zn(n+I).

The case n=3 forms an exception in which the corresponding group is not simple. It is also to be noticed that a cyclical group is a simple group, since it has no continuous self-conjugate subgroup distinct from itself. W. K. J. Killing and E. J. Cartan have separately proved that outside these four great classes there exist only five distinct types of simple groups, whose orders are 14, 52, 78, 133 and 248; thus completing the enumeration of all possible types. To prevent any misapprehension as to the bearing of these very general results, it is well to point out explicitly that there are no limitations on the parameters of a continuous group as it has, been defined above. They are to be regarded as taking in general complex values. If in the finite equations of a continuous group the imaginary See also:

symbol does not explicitly occur, the finite equations will usually define a group (in the general sense of the original definition) when both parameters and variables are limited to real values. Such a group is, in a certain sense, a continuous group; and such groups have been considered shortly by Lie (cf.

Lie-Engel, iii. 360-392), who calls them real continuous groups. To these real continuous groups the above statement as to the totality of simple groups does not apply; and indeed, in all See also:

probability, the number of types of real simple continuous groups admits of no such See also:complete enumera-- tion. The effect of See also:limitation to real transformations may be illustrated by considering the groups of'projective transformations which change x2+y2+z2—I =o and x2+y2-z2—i =o respectively into themselves. Since one of these quadrics is changed into the other by the imaginary transformation x'=x, y' =y, z'=zs/ (—I),the general continuous groups which transform the two quadrics respectively into themselves are simply isomorphic. This is not, however, the case for the real continuous groups. In fact, the second quadric has two real sets of generators; and therefore the real group which transforms it into itself has two self-conjugate subgroups, either of which leaves unchanged each of one set of generators. The first quadric having imaginary generators, no such self-conjugate subgroups can exist for the real group which transforms it into itself; and this real group is in fact simple. Among the groups isomorphic with a given continuous group there is one of special importance which is known as the See also:adjunct group. This is a homogeneous linear group in a number of variables equal to the order of the group,whose infinitesimal operations are defined by the relations Xi =MC,isxia—x,, (7 = I, 2, .. , r), i,s where c;,„ are the often-used constants, which give the combinants of the infinitesimal operations in terms of the infinitesimal operations themselves. That the r infinitesimal operations thus defined actually generate a group isomorphic with the given group is verified by forming their combinants.

It is thus found that (XPXg) =Mc55aXs. The X's, however, are not necessarily linearly independent. In fact, the sufficient condition that ~a2X, should be identically zero is that Eaiciis should vanish for all values of i and s. Hence if the equations Eaic;i,,=o for all values of i and s, have r' linearly independent solutions, only r—r' of the X's are linearly independent, and the isomorphism of the two groups is multiple. If Yl, Y2, ..., Y, are the infinitesimal operations of the given group, the equations Eaiciis=o, (s, i=1, 2,...,'r) See also:

express the condition that the operations of the cyclical group generated by EaiYi should be permutable with every operation of i the group; in other words, that they should be self-conjugate operations. In the case supposed, therefore, the given group contains a subgroup of order r' each of whose operations is self-conjugate. The adjunct group of a given group will therefore be simply isomorphic with the group, unless the latter contains self-conjugate operations; and when this is the case the order of the adjunct will be less than that of the given group by the order of the subgroup formed of the self-conjugate operations. We have been thus far mainly concerned with the abstract theory of continuous groups, in which no distinction is made be- t*een two simply isomorphic groups. We proceed to Continuous discuss the See also:classification and theory of groups when groups of the their form is regarded as essential; and this is a return "fie ofthe See also:plane, and to a more geometrical point of view. of three It is natural to begin with the projective groups, dimensional which are the simplest in form and at the same See also:time are space. of supreme importance in See also:geometry. The general See also:pro- jective group of the straight See also:line is the group of order three given by ax+b x cx+d' where the parameters are the ratios of a, b, c, d. Since X'3-X'2 x —x'1 _x3—x2 x—xl x'—x'2 x3—xl.x—x2 is an operation of the above form, the group is triply transitive. Every subgroup of order two leaves one point unchanged, and all such subgroups are conjugate.

A cyclical subgroup leaves either two distinct points or two coincident points unchanged. A subgroup which either leaves two points unchanged or interchanges them is an example of a " mixed " group. The See also:

analysis of the general projective group must obviously increase very rapidly in complexity, as the dimensions of the space to which it applies increase. This analysis has been completely carried out for the projective group of the plane, with the result of showing that there are See also:thirty distinct types of subgroup. Excluding the general group itself, every one of these leaves either a point, a line, or a conic See also:section unaltered. For space of three dimensions Lie has also carried out a similar investigation, but the results are extremely complicated. One general result of great importance at which Lie arrives in this connexion is that every projective group in space of three dimensions, other than the general group, leaves either a point, a See also:curve, a See also:surface or a linear complex unaltered. Returning now to the case of a single variable, it can be shown that any finite continuous group in one variable is either cyclical or of order two or three, and that by a suitable transformation any such group may be changed into a projective group. The See also:genesis of an infinite as distinguished from a finite continuous group may be well illustrated by considering it in the case of a single variable. The infinitesimal operations of the projective group in one variable are dx' xax, x2. . If these combined with xadx be The adjunct group. will interchange the surface-elements of space among themselves, and will change any system of x2 elements into another system of o02 elements.

A special system, i.e. a system which belongs to a point, curve or surface, will not, however, in general be changed into another special system. The necessary and sufficient condition that a special system should always be changed into a special system is that the See also:

equation dz' p'dx' q`dy' =o should be a consequence of the equation dz-pdx-qdy=o; or, in other words, that this latter equation should be invariant for the transformation. When this condition is satisfied the transformation is such as to change the surface-elements of a surface in general into surface-elements of a surface, though in particular cases they may become the surface-elements of a curve or point; and similar statements may be made with respect to a curve or point. The transformation is therefore a veritable geometrical transformation in space of three dimensions. Moreover, two special systems of surface-elements which have an See also:element in See also:common are transformed into two new special systems with an element in common. Hence two curves or surfaces which See also:touch each other are transformed into two new curves or surfaces which touch each other. It is this See also:property which leads to the transformations in question being called contact-transformations. It will be noticed that an ordinary point-transformation is always a contact-transformation, but that a contact-transformation (in space of n dimensions) is not in general a point-transformation (in space of n dimensions), though it may always be regarded as a point-transformation in space of 2n+I dimensions. Inthea.nalogous theory for space of two dimensions a line-element, defined by (x, y, p), where I : p gives the direction-cosines of the line, takes the See also:place of the surface-element ; and a transformation of x, y and p which leaves the equation dy-pdx =o unchanged transforms the oo i line-elements, which belong to a curve, into ooi line-elements which again belong to a curve; while two curves which touch are transformed into two other curves which touch. One of the simplest instances of a contact-transformation that can be given is the transformation by reciprocal polars. By this trans-formation a point P and a plane p passing through it are changed into a plane p' and a point P' upon it; i.e. the surface-element defined by P, p is changed into a definite surface-element defined by P', p'. The totality of surface-elements which belong to a (non-developable) surface is known from geometrical considerations to be changed into the totality which belongs to another (non-developable) surface.

On the other hand, the totality of the surface-elements which belong to a curve is changed into another set which belong to a developable. The analytical formulae for this transformation, when the reciprocation is effected with respect to the paraboloid x2+y2-2z=o, are x'=p, y'=q, z'=px+gy—z, p'=x, q'=y. That this is, in fact, a contact-transformation is verified directly by noticing that dz'—p'dx'—q'dy'= — d(z—px—qy) —xdp—ydq= —(dz—pdx—qdy). A second simple example is that in which every surface-element is displaced, without change of See also:

orientation, normal to itself through a constant distance E. The analytical equations in this case are easily found in the form taken as infinitesimal operations from which to generate a continuous group among the infinitesimal operations of the group, there must occur the combinant of x2dx and x2dx This is x4dx. The combinant of this and x2dx is 2x'dx and so on. Hence xrdx' where r is any See also:positive integer, is an infinitesimal operation of the group. The general infinitesimal operation of the group is therefore f(x)dx, where f(x) is an arbitrary integral See also:function of x. In the classification of the groups, projective or non-projective of two or more variables, the distinction between See also:primitive and imprimitive groups immediately presents itself. For groups of the plane the following question arises. Is there or is there not a singly-infinite See also:family of curves f(x, y) =C, where C is an arbitrary constant such that every operation of the group interchanges the curves of the family among themselves? In accordance with the previously given definition of imprimitivity, the group is called imprimitive or primitive according as such a set exists or not.

In space of three dimensions there are two possibilities; namely, there may either be a singly infinite system of surfaces F (x, y, z) --C, which are inter-changed among themselves by the operations of the group; or there may be a doubly-infinite system of curves G(x, y, z) =a, H(x, y, z)=b, which are so interchanged. In regard to primitive groups Lie has shown that any primitive group of the plane can, by a suitably chosen transformation, be transformed into one of three definite types of projective groups; and that any primitive group of space of three dimensions can be transformed into one of eight definite types, which, however, cannot all be represented as projective groups in three dimensions. The results which have been arrived at for imprimitive groups in two and three variables do not admit of any such simple statement. We shall now explain the conception of contact-transformations Contact and groups of contact-transformations. This concep- cont a t tion, like that of continuous groups, owes its origin to transf or- Lie. . From a purely analytical point of view a contact-transformation may be defined as a point-transformation in 211+I variables, z, x2, . , x", pz, • . , p" which leaves unaltered the equation dz—pidxi—p2dx2— ... —p"dx"=o. Such a definition as this, however, gives no direct See also:

clue to the geometrical properties of the transformation, nor does it explain the name given. In dealing with contact-transformations we shall restrict ourselves to space of two or of three dimensions; and it will be necessary to begin with some purely geometrical considerations. An infinitesimal surface-element in space of three dimensions is completely specified, apart from its See also:size, by its position and orientation.

If x, y, z are the co-ordinates of some one point of the element, and if p, q, -I give the ratios of the direction-cosines of its normal, x, y, z, p, q are five quantities which completely specify the element. There are, therefore, oo5 surface elements in three-dimensional space. The surface-elements of a surface form a system of o02 elements, for there are x 2 points on the surface, and at each a definite surface-element. The surface-elements of a curve form, again, a system of o02 elements, for there are x i points on the curve, and at each x i surface-elements containing the tangent to the curve at the point. Similarly the surface-elements which contain a given point clearly form a system of x 2 elements. Now each of these systems of co 2 surface-elements has the property that if (x, y, z, p, q) and (x+dx, y+dy, z+dz, p+dp, q+dq) are consecutive elements from any one of them, then dz—pdx—qdy=o. In fact, for a system of the first See also:

kind dx, dy, dz are proportional to the direction-cosines of a tangent line at a point of the surface, and p, q, —I are proportional to the direction-cosines of the normal. For a system of the second kind dx, dy, dz are proportional to the direction-cosines of a tangent to the curve, and p, q, -I give the direction-cosines of the normal to a plane touching the curve; and for a system of the third kind dx, dy, dz are zero. Now the most general way in which a system of o02 surface-elements can be given is by three independent equations between x, y, z, p and q. If these equations do not contain p, q, they determine one or more (a finite number in any case) points in space, and the system of surface-elements consists of the elements containing these points; i.e. it consists of one or more systems of the third kind. If the equations are such that two distinct equations independent of p and q can be derived from them, the points of the system of surface-elements lie on a curve. For such a system the equation dz-pdx-qdy=o will hold for each two consecutive elements only when the plane of each element touches the curve at its own point.

If the equations are such that only one equation independent of p and q can be derived from them, the points of the system of surface-elements lie on a surface. Again, for such a system the equation dz-pdx-qdy=o will hold for each two consecutive elements only when each element touches the surface at its own point. Hence, when all possible systems of m2 surface-elements in space are considered, the equation dz-pdx-qdy=o is characteristic of the three special types in which the elements belong, in the sense explained above, to a point or a curve or a surface. Let us consider now the geometrical bearing of any transformation x'=fi(x, y, z, p, q), ..., q' =Mx, y, z, p, q),of the five variables. It x' x+ Pt qt J I+p2+g2 y =y+„j +p2 +q2, +p2+g2, p' = q, q' =q. That this is a contact-transformation is seen geometrically by noticing that it changes a surface into a parallel surface. Every point is changed by it into a See also:

sphere of See also:radius t, and when t is regarded as a parameter the equations define a cyclical group of contact-transformations. The formal theory of continuous groups of contact-transformations, is, of course, in no way distinct from the formal theory of continuous groups in general. On what may be called the geometrical See also:side, the theory of groups of contact-transformations has been See also:developed with very considerable detail in the second See also:volume of Lie-Engel. To the manifold applications of the theory of continuous groups in various branches of pure and applied See also:mathematics Appticait is impossible here to refer in any detail. It must suffice to indicate a few of them very briefly. In some ttohenstheory o of the older theories a new point of view is obtained which t he t he presents the results in a fresh See also:light, and suggests the tinuous natural generalization.

As an example, the theory of groups. the invariants of a binary form may be considered. If in the form f=aox"+naix"-iy+ the variables be subjected to a homogeneous substitution x'=ax+/3y, y'=yx+Sy, (i.) and if the coefficients in the new form be represented by accenting the old coefficients, then a'o = ¢Dori+a,nan-iy+... +a"yn, a'i=aoan-i$+ai{(n-I)a"-2,By+a'- )+...+a"yn-iS, F (ii.) a' = ao/3,+arn/3"-iS+ ... +(1"S" ; and this is a homogeneous linear substitution performed on the coefficients. The totality of the substitutions, (i.), for which aS—f3y=I, constitutes a continuous group of order 3, which is generated by the two infinitesimal transformations ydx and xay. Hence with t a1 a +(n`I)a2 a +(n—2)aia n +... +au? cae cal See also:

cat oai The invariants of the binary form, i.e. those functions of the coefficients which are unaltered by all homogeneous substitutions on x, y of See also:determinant unity, are therefore identical with the functions of the coefficients which are invariant for the continuous group generated by the two infinitesimal operations last written. In other words, they are given by the common solutions of the differential equations aoa+2aiaa,+3a2aa +... =o, nal ao+(n-I)a2a¢ +(n—2)a2 +... =o. Both this result and the method by which it is arrived at are well known, but the point of view by which we pass from the transformation group of the variables to the isomorphic transformation group of the coefficients, and regard the invariants as invariants rather of the group than of the forms, is a new and a fruitful one.

The general theory of curvature of curves and surfaces may in a similar way be regarded as a theory of their invariants for the group of motions. That something more than a See also:

mere change of phraseology is here implied will be evident in dealing with minimum curves, i.e. with curves such that at every point of them dx2+dye+dz2=o. For such curves the ordinary theoryof curvature has no meaning, but they nevertheless have invariant properties in regard to the group of motions. The curvature and torsion of a curve, which are invariant for all transformations by the group of motions, are special instances of what are known as differential invariants. If n is the ox oy general infinitesimal transformation of a group of point-transformations in the plane, and if y2,... represent the successive differential coefficients of y, the infinitesimal transformation may be written in the extended form ax+'1]y+Ttl+n2aY2+ .. . where n1St, n2St, . . . are the increments of yl, y2, . . . By including a sufficient number of these variables the group must be intransitive in them, and must therefore have one or more invariants. Such invariants are known as differential invariants of the original group, being necessarily functions of the differential coefficients of the original variables. For groups of the plane it may be shown that not more than two of these differential invariants are independent, all others being formed from these by algebraical processes and differentiation. For groups of point-transformations in more than two variables there will be more than one set of differential invariants.

For instance, with three variables, one may be regarded as independent and the other two as functions of it, or two as independent and the remaining one as a function. Corresponding to these two points of view, the differential invariants for a curve or for a surface will arise. If a differential invariant of a continuous group of the plane be equated to zero, the resulting:differential equation remains unaltered when the variables undergo any transformation of the group. Conversely, if an ordinary, differential equation f(x, y, y1, y2, . . . ) =o admits the transformations of a continuous group, i.e. if the equation is unaltered when x and y undergo any transformation of the group, then f(x, y, y,, y2, . . ) or some multiple of it must be a differential invariant of the group. Hence it must be possible to find two independent differential Invariants a, p of the group, such that when these are taken as variables the differential equation takes the form F(a, a-.$, ... ) =o. This equation in a, l3 will be of See also:

lower order da da than the original equation, and in general simpler to See also:deal with. Supposing it solved in the form /3 0(a), where for a, fi their values in terms of x, y, yi, y2, ... are written, this new equation, containing arbitrary constants, is necessarily again of lower order than the original equation. The integration of the original equation is thus divided into two steps.

This will show how, in the case of an ordinary differential equation, the fact that the equation admits a continuous group of transformations maybe taken See also:

advantage of for its integration. 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 See also: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 equationhas 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 See also: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 a =o or xdy =o. 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 See also:consideration of discontinuous groups. Although groups of finite order are necessarily contained under this general See also:head, it is convenient for many reasons to deal with them separately, and it will therefore be assumed in the See also: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.

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 General. indistinguishable. The number of generating operations See also:

ing See also:opera-may be either finite or infinite, but the former case alone tions. will be here considered. Suppose then that SI, S2, ..., S„ 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 SaSb . . . Sa, or Z, where a, b, ..., d are chosen from I, 2, ..., n, and a, $, ..., S are any positive or negative integers. It may be assumed that no two successive suffixes in E are the same, for if b =a, then SaSb may be replaced by Sa+P. If there are no relations connecting the generating operations and the identical operation, every distinct symbol E represents a distinct operation of the group. For If = Fl, or SaSb ... Sa = SaiSb l ...Sal, then Shc' ... Sb A'Sal'SaSb ... Sd =I; and unless a=ai,b=b1,..

,a=al,i9=t1,... ,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 SI, S2, ..., S,,, the set ZH, and no others, reduce to the same operation B when the conditions T1= I, T2 =I, . . . 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 See also:

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 See also: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 specified analytically by a system of equations of the form x'1=fe(x', x2, ..., x„ ; al, a2, ..., a.), (s = I, 2, ..., n), and the different operations of the group will be given by different sets of values of the parameters a1, a2, ... , a,. No one of these parameters is susceptible of continuous See also:

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 See also:differences x'1-x1, x'2-x2, ..., x',. x, 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 the same limitations on a, ,B, y, S the totality of the substitutions (ii.) forms a simply isomorphic continuous group of order 3, which is generated by the two infinitesimal transformations aaal+2alaa,+3a3+... +na„_lc , and Properly and lm-properly discontinuous groups. 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' =cxax+b +d' where a, b, c, d are integers satisfying the relation ad-bc = I, 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'.=.f,(xi, x2, ..., x,,; al, a2, ..., at), (s=l, 2, ..., n), are the finite equations of a continuous group, and if C with See also:

para- meters co c2, ... , c,. is the operation which results from carrying out A and B with corresponding parameters in See also: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. Linear A most important class of discontinuous groups are those discon- that arise in this way from the general linear continuous tinuous group in a given set of variables. For n variables the groups, finite equations of this continuous group are x'.=aaxl+a.2x2+ . . • +a=x,,, (s=I, 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+byy' =cx+dy, where a, b, c, d are integers such that ad-bc= i. To every operation of this group there corresponds an operation of the set defined by az+b 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 See also:

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 geo- metrical 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 See also:im- properly discontinuous; but when the generating opera- tions are suitably chosen, the group may be properly discontinuous. In the latter case the group may be represented in a graphical form by the See also: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 See also:

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 See also:part. By See also: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 See also:round B if the See also: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 13C, 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 them-selves by any even number of inversions. Hence the operations consisting of an even number of inversions form a group by them-selves. For this group the See also:quadrilateral formed by ABC andA'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 See also: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 trans-formed by any even number of inversions, then z' and z are connected by a linear relation z' ='YZaz+R+S' where a, ,B, y, S 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'=Yizaiz++SRi r' z'=72z+32, a2z272z+32, 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=o, x2+y2_ I =o, x2+y2-}-2x=o, the generating operations are found to be z'=z+i, z'= -z'; and the group is that consisting of all trans- formations of the form z'=azz+b +—d' where ad-bc=l, a, b, c, d being cz 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 See also:

index of the group in question. The triangle ABC from which the above geometrical construction started may be replaced by a See also: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. See also: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 with in the previous paragraphs arise directly from geometrical considerations. As a final example we shall refer briefly See also:

Croup 01 to a class of groups whose origin is essentially analytical. a linear Let differen- tial equation. Discontinuous groups arising from geometrical operations. de-1y ... 4-P,-lax+Psy be a linear differential equation, the coefficients in which are rational functions of x, and let y2, ..., yn be a linearly independent set of integrals of the equation. In the neighbourhood of a finite value xo of x, which is not a singularity of any of the coefficients in the equation, these integrals are ordinary See also:power-series in x-xo. If the analytical continuations of yl, y2, ..., y„ be formed for any closed path starting from and returning to xo, the final values arrived at when xo 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'i, y'2, ..., Y. be the new integrals arrived at. Since in the neighbourhood of xo every integral can be represented linearly in terms of y2, ... , y" , there must be a system of equations y'1=any' +a12y2+ .

. . +alnyn, yz = a2lyl +a22y2+ . . . +a2n_vn, Y'. = anlyi+a.,1y2+ ... +a„nyn, where the a's are constants, expressing the new integrals in terms of the original ones. To each closed path described by xo there therefore corresponds a definite linear substitution performed on the y's. Further, if Si and S2 are the substitutions that correspond to two closed paths Li and L2, then to any closed path which can be See also:

con- tinuously deformed, without See also:crossing a singular point, into Li followed by L2, there corresponds the substitution See also:S1S2. Let L1, L2, ..., Lr be arbitrarily chosen closed paths starting from and return- ing 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 Si, S2, S3, ...

, Sr 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„ 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 Si, S2, S3, ... , SN denote the operations of a group G of finite order N, Si being the identical operation. The tableau S1, S2, S3, .

End of Article: GROUPS, THEORY

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