the constants cijs and dijs are the same for all values of i, j and s, the two groups are simply isomorphic, Xs and Ys being corresponding infinitesimal operations.
Two continuous groups of order r, whose infinitesimal operations obey the same system of equations (iii.), may be of very different form; for instance, the number of variables for the one may be different from that for the other. They are, however, said to be of the same type, in the sense that the laws according to which their operations combine are the same for both.
The problem of determining all distinct types of groups of order r is then contained in the purely algebraical problem of finding all the systems of r3 quantities cijs which satisfy the relations
cijt + cijt = 0,
Σs cijs cskt + cjks csit + ckis csjt = 0.
for all values of i, j, k and t. To two distinct solutions of the algebraical problem, however, two distinct types of group will not necessarily correspond. In fact, X1, X2, ..., Xr may be replaced by any r independent linear functions of themselves, and the c’s will then be transformed by a linear substitution containing r2 independent parameters. This, however, does not alter the type of group considered.
For a single parameter there is, of course, only one type of group, which has been called cyclical.
For a group of order two there is a single relation
(X1X2) = αX1 + βX2.
If α and β are not both zero, let α be finite. The relation may then be written (αX1 + βX2, α−1X2) = αX1 + βX2. Hence if αX1 + βX2 = X′1, and α−1X2 = X′2, then (X′1X′2) = X′1. There are, therefore, just two types of group of order two, the one given by the relation last written, and the other by (X1X2) = 0.
Lie has determined all distinct types of continuous groups of orders three or four; and all types of non-integrable groups (a term which will be explained immediately) of orders five and six (cf. Lie-Engel, iii. 713-744).