Part 2
As a second example the group of motions in three-dimensional space will be considered. The totality of motions, i.e. of space displacements which leave the distance of every pair of points unaltered, obviously constitutes a set of operations which satisfies the group definition. From the elements of kinematics it is known that every motion is either (i.) a translation which leaves no point unaltered, but changes each of a set of parallel lines into itself; or (ii.) a rotation which leaves every point of one line unaltered and changes every other point and line; or (iii.) a twist which leaves no point and only one line (its axis) unaltered, and may be regarded as a translation along, combined with a rotation round, the axis. Let S be any motion consisting of a translation l along and a rotation a round a line AB, and let T be any other motion. There is some line CD into which T changes AB; and therefore T^(-1)ST leaves CD unchanged. Moreover, T^(-1)ST clearly effects the same translation along and rotation round CD that S effects for AB. Two motions, therefore, are conjugate if and only if the amplitudes of their translation and rotation components are respectively equal. In particular, all translations of equal amplitude are conjugate, as also are all rotations of equal amplitude. Any two translations are permutable with each other, and give when combined another translation. The totality of translations constitutes, therefore, a subgroup of the general group of motions; and this subgroup is a self-conjugate subgroup, since a translation is always conjugate to a translation.
All the points of space constitute a set of objects which are interchanged among themselves by all operations of the group of motions. So also do all the lines of space and all the planes. In respect of each of these sets the group is simply transitive. In fact, there is an infinite number of motions which change a point A to A', but no motion can change A and B to A' and B' respectively unless the distance AB is equal to the distance A'B'.
The totality of motions which leave a point A unchanged forms a subgroup. It is clearly constituted of all possible rotations about all possible axes through A, and is known as the group of rotations about a point. Every motion can be represented as a rotation about some axis through A followed by a translation. Hence if G is the group of motions and H the group of translations, G/H is simply isomorphic with the group of rotations about a point.
The totality of the motions which bring a given solid to congruence with itself again constitutes a subgroup of the group of motions. This will in general be the trivial subgroup formed of the identical operation above, but may in the case of a symmetrical body be more extensive. For a sphere or a right circular cylinder the subgroups are those that leave the centre and the axis respectively unaltered. For a solid bounded by plane faces the subgroup is clearly one of finite order. In particular, to each of the regular solids there corresponds such a group. That for the tetrahedron has 12 for its order, for the cube (or octahedron) 24, and for the icosahedron (or dodecahedron) 60.
The determination of a particular operation of the group of motions involves six distinct measurements; namely, four to give the axis of the twist, one for the magnitude of the translation along the axis, and one for the magnitude of the rotation about it. Each of the six quantities involved may have any value whatever, and the group of motions is therefore a continuous group. On the other hand, a subgroup of the group of motions which leaves a line or a plane unaltered is a mixed group.
We shall now discuss (i.) continuous groups, (ii.) discontinuous groups whose order is not finite, and (iii.) groups of finite order. For proofs of the statements, and the general theorems, the reader is referred to the bibliography.
_Continuous Groups._
The determination of a particular operation of a given continuous group depends on assigning special values to each one of a set of parameters which are capable of continuous variation. The first distinction regards the number of these parameters. If this number is finite, the group is called a _finite_ continuous group; if infinite, it is called an _infinite_ continuous group. In the latter case arbitrary functions must appear in the equations defining the operations of the group when these are reduced to an analytical form. The theory of infinite continuous groups is not yet so completely developed as that of finite continuous groups. The latter theory will mainly occupy us here.
Sophus Lie, to whom the foundation and a great part of the development of the theory of continuous groups are due, undoubtedly approached the subject from a geometrical standpoint. His conception of an operation is to regard it as a geometrical transformation, by means of which each point of (n-dimensional) space is changed into some other definite point.
The representation of such a transformation in analytical form involves a system of equations,
x'_s = [f]_s(x1, x2, ..., x_n), (s = 1, 2, ..., n),
expressing x'1, x'2, ..., x'_n, the co-ordinates of the transformed point in terms of x1, x2, ..., x_n, the co-ordinates of the original point. In these equations the functions [f]_s are analytical functions of their arguments. Within a properly limited region they must be one-valued, and the equations must admit a unique solution with respect to x1, x2, ..., x_n, since the operation would not otherwise be a definite one.
From this point of view the operations of a continuous group, which depends on a set of r parameters, will be defined analytically by a system of equations of the form
x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n), (i.)
where a1, a2, ..., a_r represent the parameters. If this operation be represented by A, and that in which b1, b2, ..., b_r are the parameters by B, then the operation AB is represented by the elimination (assumed to be possible) of x'1, x'2, ..., x'_n between the equations (i.) and the equations
x"_s = [f]_s(x'1, x'2, ..., x'_n; b1, b2, ..., b_r), (s = 1, 2, ..., n).
Since AB belongs to the group, the result of the elimination must be
x"_s = [f]_s(x1, x2, ..., x_n; c1, c2, ..., c_r),
where c1, c2, ..., c_r represent another definite set of values of the parameters. Moreover, since A^(-1) belongs to the group, the result of solving equations (i.) with respect to x1, x2, ..., x_n must be
x_s = [f]_s(x'1, x'2, ..., x'_n; d1, d2, ..., d_r), (s = 1, 2, ..., n).
Conversely, if equations (i.) are such that these two conditions are satisfied, they do in fact define a finite continuous group.
Infinitesimal operation of a continuous group.
It will be assumed that the r parameters which enter in equations (i.) are independent, i.e. that it is impossible to choose r' (< r) quantities in terms of which a1, a2, ..., a_r can be expressed. Where this is the case the group will be spoken of as a "group of order r." Lie uses the term "_r-gliedrige Gruppe_." It is to be noticed that the word order is used in quite a different sense from that given to it in connexion with groups of finite order.
In regard to equations (i.), which define the general operation of the group, it is to be noticed that, since the group contains the identical operation, these equations must for some definite set of values of the parameters reduce to x'1 = x1, x'2 = x2, ..., x'_n = x_n. This set of values may, without loss of generality, be assumed to be simultaneous zero values. For if i1, i2, ..., i_r be the values of the parameters which give the identical operation, and if we write
a_s = i_s + a, (s = 1, 2, ..., r),
then zero values of the new parameters a1, a2, ..., a_r give the identical operation.
To infinitesimal values of the parameters, thus chosen, will correspond operations which cause an infinitesimal change in each of the variables. These are called infinitesimal operations. The most general infinitesimal operation of the group is that given by the system
[Pd][f]_s [Pd][f]_s [Pd][f]_s x'_s - x_s = [delta]x_s = --------- [delta]a1 + --------- [delta]a2 + ... + --------- [delta]a_r, (s = 1, 2, ..., n), [Pd]a1 [Pd]a2 [Pd]a_r
where, in [Pd][f]_s/[Pd]a_i, zero values of the parameters are to be taken. Since a1, a2, ..., a_r are independent, the ratios of [delta]a1, [delta]a2, ..., [delta]a_r are arbitrary. Hence the most general infinitesimal operation of the group may be written in the form
/ [Pd][f]_s [Pd][f]_s [Pd][f]_s\ [delta]x_s = ( e1--------- + e2--------- + ... + e_r--------- ) [delta]t, (s = 1, 2, ..., n), \ [Pd]a1 [Pd]a2 [Pd]a_r /
where e1, e2, ..., e_r are arbitrary constants, and [delta]t is an infinitesimal.
If F(x1, x2, ..., x_n) is any function of the variables, and if an infinitesimal operation of the group be carried out on the variables in F, the resulting increment of F will be
[Pd]F [Pd]F [Pd]F ------[delta]x1 + ------[delta]x2 + ... + -------[delta]x_n. [Pd]x1 [Pd]x2 [Pd]x_n
If the differential operator
[Pd][f]1 [Pd] [Pd][f]2 [Pd] [Pd][f]_n [Pd] -------- ------ + -------- ------ + ... + --------- ------- [Pd]a_i [Pd]x1 [Pd]a_i [Pd]x2 [Pd]a_i [Pd]x_n
be represented by X_i, (i = 1, 2, ..., r), then the increment of F is given by
(e1X1 + e2X2 + ... + e_rX_r)F[delta]t.
When the equations (i.) defining the general operation of the group are given, the coefficients [Pd][f]_s/[Pd]a_i, which enter in these differential operators are functions of the variables which can be directly calculated.
The differential operator e1X1 + e2X2 + ... + e_rX_r 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 (1 + [delta]tX)F is the result of carrying out the infinitesimal operation on F; and by putting x1, x2, ..., x_n 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 e1, e2, ..., e_r have assigned values. Then the result of repeating the particular infinitesimal operation e1X1 + e2X2 + ... + e_rX_r or X an 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 [delta]t is the infinitesimal already introduced, then
dF d^2F -- = X.F, ---- = X.X.F, ... dt dt^2
Hence
dF t^2 d^2F F' = F + t-- + --- ---- + ... dt 1.2 dt^2
t^2 = F + tX.F + --- X.X.F + ... 1.2
It must, of course, be understood that in this analytical representation of the effect of the finite operation on F it is implied that t is taken sufficiently small to ensure the convergence of the (in general) infinite series.
When x1, x2, ... are written in turn for F, the system of equations
t^2 x'_s = (1 + tX + --- X.X + ...)x_s, (s = 1, 2, ..., n) (ii.) 1.2
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 case results immediately from noticing that the result of eliminating F' between
t^2 F' = F + tX.F + --- X.X.F + ... 1.2
and
t'^2 F" = F' + t'X.F' + ---- X.X.F' + ... 1.2
is
(t + t')^2 F" = F + (t + t') X.F + ---------- X.X.F + ... 1.2
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 e1, e2, ..., e_r may be. Hence if e1t, e2t, ..., e_rt be replaced by a1, a2, ..., a_r the equations (ii.) represent a set of operations, depending on r parameters and belonging to the group. They must therefore be a form of the general equations for any operation of the group, and are 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 ordinary differential equations. As a very simple example we may consider the case in which the infinitesimal operation is given by X = x^2[Pd]/[Pd]x, so that there is only a single variable. The relation between x' and t is given by dx'/dt = x'^2, with the condition that x' = x when t = 0. This gives at once x' = x/(1 - tx), which might also be obtained by the direct use of (ii.).
Relations between the infinitesimal operations of a finite continuous group.
When the finite equations (i.) of a continuous group of order r are known, it has now been seen that the differential operator which defines the most general infinitesimal operation of the group can be directly constructed, and that it contains r arbitrary constants. This is equivalent to saying that the group contains r linearly independent infinitesimal operations; and that the most general infinitesimal operation is obtained by combining these linearly with constant coefficients. Moreover, when any r independent infinitesimal operations of the group are known, it has 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 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 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 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 (Lie uses the expression _Klammerausdruck_) and is denoted by (XY). If X, Y, Z are any three linear differential operators the identity (known as 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 X1, X2, ..., X_r 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
__k=r (X_iX_j) = \ c_(ijk)X_k, /__k=1
where the c's are constants. Moreover, from Jacobi's identity and the identity (XY) + (YX) = 0 it follows that the c's are subject to the relations
c_(ijt) + c_(jit) = 0, \ | and > | [Sigma][s](c_(jks)c_(ist) + c_(kis)c_(jst) + c_(ijs)c_(kst)) = 0 / (iii.)
for all values of i, j, k and t.
Determination of the distinct types of continuous groups of a given order.
The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in order that X1, X2, ..., X_r may generate, as infinitesimal operations, a continuous group of order r, are also sufficient.
For the proof of this fundamental theorem see Lie's works (cf. Lie-Engel, i. chap. 9; iii. chap. 25).
If two continuous groups of order r are such that, for each, a set of linearly independent infinitesimal operations X1, X2, ..., X_r and Y1, Y2, ..., Y_r can be chosen, so that in the relations
(X_iX_j) = [Sigma]c_(ijs)X_s, (Y_iY_j) = [Sigma]d_(ijs)Y_s,
the constants c_(ijs) and d_(ijs) are the same for all values of i, j and s, the two groups are simply isomorphic, X_s and Y_s 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 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 r^3 quantities c_(ijs) which satisfy the relations
c_(ijt) + c_(ijt) = 0,
[Sigma] [c_(ijs)c_(skt) + c_(jks)c_(sit) + c_(kis)c_(sjt)] = 0. s
for all values of i, j, k and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, X1, X2, ..., X_r may be replaced by any r independent linear functions of themselves, and the c's will then be transformed by a linear substitution containing r^2 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) = [alpha]X1 + [beta]X2.
If [alpha] and [beta] are not both zero, let [alpha] be finite. The relation may then be written ([alpha]X1 + [beta]X2, [alpha]^(-1)X2) = [alpha]X1 + [beta]X2. Hence if [alpha]X1 + [beta]X2 = X'1, and [alpha]^(-1)X2 = X'2, then (X'1X'2) = X'1. 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 term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744).
Self-conjugate subgroups. Integrable groups.
A problem of fundamental importance in connexion with any given continuous group is the determination of the self-conjugate subgroups which it contains. If X is an infinitesimal operation of a group, and Y any other, the general form of the infinitesimal operations which are conjugate to X is
t^2 X + t(XY) + --- ((XY)Y) + .... 1.2
Any subgroup which contains all the operations conjugate to X must therefore contain all infinitesimal operations (XY), ((XY)Y), ..., where for Y each infinitesimal operation of the group is taken in turn. Hence if X'1, X'2, ..., X'_s 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
__e=1 (X'_iY) = \ a_(ie)X'_e, (i = 1, 2, ..., s) /__e=s
must be satisfied. Conversely, if such a set of relations is satisfied, X'1, X'2, ..., X'_s 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,
((X_iX_j)X_k) = [Sigma] c_(ijs)(X_sX_k). s
Of the 1/2r(r - 1) 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 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, G1 the derived group of G, G2 that of G1, and so on, the series of groups G, G1, G2, ... will terminate either with the identical operation or with a perfect group; for the order of G_(s+1) is less than that of G_s unless G_s 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 X1, X2, ..., X_r which generate the group may be chosen so that X1, X2, ..., X_(r1), (r1 < r) generate the first derived group, X1, X2, ..., X_(r2), (r2 < r1) the second derived group, and so on. When they are so chosen the constants c_(ijs) are clearly such that if r_p < i <= r_(p+1), r_q < j <= r_(q+1), p >= q, then c_(ijs) vanishes unless s <= r_(p+1).
In particular the generating operations may be chosen so that c_(ijs) vanishes unless s is equal to or less than the smaller of the two numbers i, j; and conversely, if the c's satisfy these relations, the group is integrable.
Simple groups.
A simple group, as already defined, is one which has no self-conjugate subgroup. It is a remarkable fact that the determination of all distinct types of simple continuous groups has been made, for in the case of discontinuous groups and groups of finite order this is far from being the case. Lie has demonstrated the existence of four 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
a_s, _1x1 + a_s, _2x2 + ... + a_s, _nx_n + a_(s, n + 1) x'_s = --------------------------------------------------------, (s = 1, 2, ..., n), a_(n+1), _1x1 + a_(n+1), _2x2 + ... + a_(n+1), _nx_n + 1
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-special linear complex in space of 2n - 1 dimensions with itself. The order of this group is n(2n + 1).
(iii.) and (iv.) The groups simply isomorphic with the totality of the projective transformations which change a quadric of non-vanishing discriminant into itself. These fall into two distinct classes of types according as n is even or odd. In either case the order is 1/2n(n + 1). 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.