We have been thus far mainly concerned with the abstract theory of continuous groups, in which no distinction is made between Continuous groups of the line of the plane, and of three-dimensional space. 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 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 d/dx, x(d/dx), x2(d/dx). If these combined with x3(d/dx) be taken as infinitesimal operations from which to generate a continuous group among the infinitesimal operations of the group, there must occur the combinant of x2(d/dx) and x3(d/dx). This is x4(d/dx). The combinant of this and x2(d/dx) is 2x5(d/dx) and so on. Hence xr(d/dx), where r is any positive integer, is an infinitesimal operation of the group. The general infinitesimal operation of the group is therefore ƒ(x)(d/dx), where ƒ(x) is an arbitrary integral function of x.
In the classification of the groups, projective or non-projective of two or more variables, the distinction between primitive and imprimitive groups immediately presents itself. For groups of the plane the following question arises. Is there or is there not a singly-infinite family of curves ƒ(x, y) = C, where C is an arbitrary constant such that every operation of the group interchanges the curves of the family among themselves? In accordance with the previously given definition of imprimitivity, the group is called imprimitive or primitive according as such a set exists or not. In space of three dimensions there are two possibilities; namely, there may either be a singly infinite system of surfaces F(x, y, z) = C, which are interchanged among themselves by the operations of the group; or there may be a doubly-infinite system of curves G(x, y, z) = a, H(x, y, z) = b, which are so interchanged.