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).