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 Simple groups. 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
| 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 ½n(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