We go on now to the consideration of discontinuous groups. Although groups of finite order are necessarily contained under this general head, it is convenient for many reasons to deal with them separately, and it will therefore be assumed in the present section that the number of operations in the group is not finite. Many large classes of discontinuous groups have formed the subject of detailed investigation, but a general formal theory of discontinuous groups can hardly be said to exist as yet. It will thus be obvious that in considering discontinuous groups it is necessary to proceed on different lines from those followed with continuous groups, and in fact to deal with the subject almost entirely by way of example. Generating operations. The consideration of a discontinuous group as arising from a set of independent generating operations suggests a purely abstract point of view in which any two simply isomorphic groups are indistinguishable. The number of generating operations may be either finite or infinite, but the former case alone will be here considered. Suppose then that S1, S2, ..., S_n is a set of independent operations from which a group G is generated. The general operation of the group will be represented by the symbol S_a^[alpha]S_b^[beta] ... S_d^[delta], or [Sigma], where a, b, ..., d are chosen from 1, 2, ..., n, and [alpha], [beta], ..., [delta] are any positive or negative integers. It may be assumed that no two successive suffixes in [Sigma] are the same, for if b = a, then S_a^[alpha]S_b^[beta] may be replaced by S_a^([alpha] +[beta]). If there are no relations connecting the generating operations and the identical operation, every distinct symbol [Sigma] represents a distinct operation of the group. For if [Sigma] = [Sigma]1, or S_a^[alpha] S_b^[beta] ... S_d^[delta] = S_(a1)^([alpha]1) S_(b1)^([beta]1) ... S_(d1)^([delta]1), then S_(d1)^(-[delta]1) ... S_(b1)^(-[beta]1) S_(a1)^(-[alpha]1) S_a^[alpha] S_b^[beta] ... S_d^[delta] = 1; and unless a = a1, b = b1, ..., [alpha] = [alpha]1, [beta] = [beta]1, ..., this is a relation connecting the generating operations.