If x1 and x2 are any pair of distinct rational numbers, and y1 and y2 any other pair, there is just one operation of the group which changes x1 and x2 into y1 and y2 respectively. For the equations y1 = ax1 + b, y1 = ax2 + b determine a and b uniquely. The group is therefore doubly transitive in respect of the set of rational numbers. If H is the subgroup that leaves unchanged a given rational number x1, and S an operation changing x1 into x2, then every operation of S−1HS leaves x2 unchanged. The subgroups, each of which leaves a single rational number unchanged, therefore form a single conjugate set. The group of multiplications leaves zero unchanged; and, as has been seen, this is conjugate with the subgroup formed of all operations x′ − d = a(x − d), where d is a given rational number. This subgroup leaves d unchanged.
The group of multiplications is clearly generated by the operations x′ = px, where for p negative unity and each prime is taken in turn. Every addition is obtained on transforming x′ = x + 1 by the different operations of the group of multiplications. Hence x′ = x + 1, and x′ = px, (p = −1, 3, 5, 7, . . .), form a set of independent generating operations of the group. It is a discontinuous group.
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−1ST leaves CD unchanged. Moreover, T−1ST 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 = ƒs (x1, x2, . . ., xn), (s = 1, 2, . . ., n),
expressing x′1, x′2, . . ., x′n, the co-ordinates of the transformed point in terms of x1, x2, . . ., xn, the co-ordinates of the original point. In these equations the functions ƒs 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, . . ., xn, 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 = ƒs(x1, x2, . . ., xn; a1, a2, . . ., ar), (s = 1, 2, . . ., n), | (i.) |
where a1, a2, . . ., ar represent the parameters. If this operation be
represented by A, and that in which b1, b2, . . ., br 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 = ƒs (x′1, x′2, . . ., x′n; b1, b2, . . ., br), (s = 1, 2, . . ., n).
Since AB belongs to the group, the result of the elimination must be
x″s = ƒs (x1, x2, . . ., xn; c1, c2, . . ., cr),
where c1, c2, . . ., cr 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, . . ., xn must be
xs = ƒs (x′1, x′2, . . ., x′n; d1, d2, . . ., dr), (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.
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, . . ., ar can be expressed. Where this is the case the group will be spoken of as a “group of order r.” Lie uses the Infinitesimal operation of a continuous group.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 = xn. This set of values may, without loss of generality, be assumed to be simultaneous zero values. For if i1, i2, . . ., ir be the values of the parameters which give the identical operation, and if we write
as = is + a, (s = 1, 2, . . ., r ),
then zero values of the new parameters a1, a2, . . ., ar 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
| x′s − xs = δxs = | ∂ƒs | δa1 + | ∂ƒs | δa2 + . . . + | ∂ƒs | δar, (s = 1, 2, . . ., n), |
| ∂a1 | ∂a2 | ∂ar |
where, in ∂ƒs /∂ai, zero values of the parameters are to be taken. Since a1, a2, . . ., ar are independent, the ratios of δa1, δa2, . . ., δar are arbitrary. Hence the most general infinitesimal operation of the group may be written in the form
| δxs = ( e1 | ∂ƒs | + e2 | ∂ƒs | + . . . + er | ∂ƒs | ) δt, (s = 1, 2, . . ., n), |
| ∂a1 | ∂a2 | ∂ar |
where e1, e2, . . ., er are arbitrary constants, and δt is an infinitesimal.
If F(x1, x2, . . ., xn) 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
| ∂F | δx1 + | ∂F | δx2 + . . . + | ∂F | δxn. |
| ∂x1 | ∂x2 | ∂xn |
If the differential operator
| ∂ƒ1 | ∂ | + | ∂ƒ2 | ∂ | + . . . + | ∂ƒn | ∂ | |||
| ∂ai | ∂x1 | ∂ai | ∂x2 | ∂ai | ∂xn |
