A problem of fundamental importance in connexion with any given Self-conjugate subgroups. Integrable groups. 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
| X + t(XY) + | t2 | ((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
(X′iY) = Σe=se=1 aie X′e, (i = 1, 2, ..., 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,
((XiXj) Xk) = Σs cijs (XsXk).
Of the ½r(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.