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,
Of the 12r (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 Gs+1 is less than that of Gs unless Gs 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, ..., Xr which generate the group may be chosen so that X1, X2, ..., Xr1, (r1 < r) generate the first derived group, X1, X2, ..., Xr2, (r2 < r1) the second derived group, and so on. When they are so chosen the constants cijs are clearly such that if rp < i < rp+1, rq < j < rq+1, p > q, then cijs vanishes unless s < rp+1.
In particular the generating operations may be chosen so that cijs 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.
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 hasSimple groups. 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
| x′s = | as, 1x1 + as, 2x2 + ... + as, nxn + as, n + 1 | , (s = 1, 2, ..., n), |
| an+1, 1x1 + an+1, 2x2 + ... + an+1, nxn + 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 12n(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.
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 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 probability, the number of types of real simple continuous groups admits of no such complete enumeration. The effect of limitation to real transformations may be illustrated by considering the groups of projective transformations which change
x2 + y2 + z2 − 1 = 0 and x2 + y2 − z2 − 1 = 0
respectively into themselves. Since one of these quadrics is changed into the other by the imaginary transformation
x′ = x, y ′ = y, z′ = z√ (−1),
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 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 relationsThe adjunct group.
| Xi = Σi, s cijs xi | ∂ | , (j = 1, 2, ..., r), |
| ∂xs |
where cijs 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 (XpXq) = Σs cpqsXs. The X’s, however, are not necessarily linearly independent. In fact, the sufficient condition that Σj ajXj should be identically zero is that Σj ajcijs should vanish for all values of i and s. Hence if the equations Σj ajcijs = 0 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 Y1, Y2, ..., Yr are the infinitesimal operations of the given group, the equations
express the condition that the operations of the cyclical group generated by Σj ajYi should be permutable with every operation of 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 between two simply isomorphic groups. We proceed to discuss the classification and theory of groups when their form is regarded as essential; and this is a return Continuous groups of the line of the plane, and of three-dimensional space. to a more geometrical point of view.
It is natural to begin with the projective groups, which are the simplest in form and at the same time are of supreme importance in geometry. The general projective group of the straight line is the group of order three given by
| x′ = | ax + b | , |
| cx + d ′ |
where the parameters are the ratios of a, b, c, d. Since
| x′3 − x′2 | · | x′ − x′1 | = | x3 − x2 | · | x − x1 |
| x′3 − x′1 | x′ − x′2 | x3 − x1 | 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 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 thirty distinct types of subgroup. Excluding the general group itself, every one of these leaves either a point, a line, or a conic 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 curve, a 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 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 ddx, xddx, x2ddx. If these combined with x3ddx be
