Suppose now that T1, T2, ... are operations of G, and that H is that self-conjugate subgroup of G which is generated by T1, T2, ... and the operations conjugate to them. Then, of the operations that can be formed from S1, S2, ..., S_n, the set [Sigma]H, and no others, reduce to the same operation [Sigma] when the conditions T1 = 1, T2 = 1, ... are satisfied by the generating operations. Hence the group which is generated by the given operations, when subjected to the conditions just written, is simply isomorphic with the factor-group G/H. Moreover, this is obviously true even when the conditions are such that the generating operations are no longer independent. Hence any discontinuous group may be defined abstractly, that is, in regard to the laws of combination of its operations apart from their actual form, by a set of generating operations and a system of relations connecting them. Conversely, when such a set of operations and system of relations are given arbitrarily they define in abstract form a single discontinuous group. It may, of course, happen that the group so defined is a group of finite order, or that it reduces to the identical operation only; but in regard to the general statement these will be particular and exceptional cases. Properly and improperly discontinuous groups. An operation of a discontinuous group must necessarily be specified analytically by a system of equations of the form x'_s = [f]_s(x1, x2, ..., x_n; a1, a2, ..., a_r), (s = 1, 2, ..., n),