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, ..., Sn, the set ΣH, and no others, reduce to the same operation Σ 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.
An operation of a discontinuous group must necessarily be specified Properly and improperly discontinuous groups. analytically by a system of equations of the form
x′s = ƒs (x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),
and the different operations of the group will be given by different sets of values of the parameters a1, a2, ..., ar. No one of these parameters is susceptible of continuous variations, but at least one must be capable of taking a number of values which is not finite, if the group is not one of finite order. Among the sets of values of the parameters there must be one which gives the identical transformation. No other transformation makes each of the differences x′1 − x1, x′2 − x2, ..., x′n − xn vanish. Let d be an arbitrary assigned positive quantity. Then if a transformation of the group can be found such that the modulus of each of these differences is less than d when the variables have arbitrary values within an assigned range of variation, however small d may be chosen, the group is said to be improperly discontinuous. In the contrary case the group is called properly discontinuous. The range within which the variables are allowed to vary may clearly affect the question whether a given group is properly or improperly discontinuous. For instance, the group defined by the equation x′ = ax + b, where a and b are any rational numbers, is improperly discontinuous; and the group defined by x′ = x + a, where a is an integer, is properly discontinuous, whatever the range of the variable. On the other hand, the group, to be later considered, defined by the equation x′ = (ax + b) / (cx + d), where a, b, c, d are integers satisfying the relation ad − bc = 1, is properly discontinuous when x may take any complex value, and improperly discontinuous when the range of x is limited to real values.
Among the discontinuous groups that occur in analysis, a large number may be regarded as arising by imposing limitations on the range of variation of the parameters of continuous groups. If
x′s = ƒs (x1, x2, ..., xn; a1, a2, ..., ar), (s = 1, 2, ..., n),
are the finite equations of a continuous group, and if C with parameters c1, c2, ..., cr is the operation which results from carrying out A and B with corresponding parameters in succession, then the c’s are determined uniquely by the a’s and the b’s. If the c’s are rational functions of the a’s and b’s, and if the a’s and b’s are arbitrary rational numbers of a given corpus (see [Number]), the c’s will be rational numbers of the same corpus. If the c’s are rational integral functions of the a’s and b’s, and the latter are arbitrarily chosen integers of a corpus, then the c’s are integers of the same corpus. Hence in the first case the above equations, when the a’s are limited to be rational numbers of a given corpus, will define a discontinuous group; and in the second case they will define such a group when Linear discontinuous groups. the a’s are further limited to be integers of the corpus. A most important class of discontinuous groups are those that arise in this way from the general linear continuous group in a given set of variables. For n variables the finite equations of this continuous group are
x′s = as1x1 + as2x2 + ... + asnxn, (s = 1, 2, ..., n),
where the determinant of the a’s must not be zero. In this case the c’s are clearly integral lineo-linear functions of the a’s and b’s. Moreover, the determinant of the c’s is the product of the determinant of the a’s and the determinant of the b’s. Hence equations (ii.), where the parameters are restricted to be integers of a given corpus, define a discontinuous group; and if the determinant of the coefficients is limited to the value unity, they define a discontinuous group which is a (self-conjugate) subgroup of the previous one.
The simplest case which thus presents itself is that in which there are two variables while the coefficients are rational integers. This is the group defined by the equations