(n^m - 1)(n^m - n) ... (n^m - n^(m-1)). The totality of the operations of the linear homogeneous group for which the determinant of the coefficients is congruent to unity forms a subgroup. Other subgroups arise by considering those operations which leave a function of the variables unchanged (mod. n). All such subgroups are known as linear homogeneous groups. When the ratios only of the variables are considered, there arises a _linear fractional_ group, with which the corresponding linear homogeneous group is isomorphic. Thus, if p is a prime the totality of the congruences az + b z' [equiv] ------, ad - bc [/=] 0, (mod. p) cz + d constitutes a group of order p(p^2 - 1). This class of groups for various values of p is almost the only one which has been as yet exhaustively analysed. For all values of p except 3 it contains a simple self-conjugate subgroup of index 2. A great extension of the theory of linear homogeneous groups has been made in recent years by considering systems of congruences of the form x'_r [equiv] a_(r1)x1 + a_(r2)x2 + ... + a_(rm)x_m, (r = 1, 2, ..., m), in which the coefficients a_(rs), are integral functions with real integral coefficients of a root of an irreducible congruence to a prime modulus. Such a system of congruences is obviously limited in numbers and defines a group which contains as a subgroup the group defined by the same congruences with ordinary integral coefficients. Applications. The chief application of the theory of groups of finite order is to the theory of algebraic equations. The analogy of equations of the second, third and fourth degrees would give rise to the expectation that a root of an equation of any finite degree could be expressed in terms of the coefficients by a finite number of the operations of addition, subtraction, multiplication, division, and the extraction of roots; in other words, that the equation could be solved by radicals. This, however, as proved by Abel and Galois, is not the case: an equation of a higher degree than the fourth in general defines an algebraic irrationality which cannot be expressed by means of radicals, and the cases in which such an equation can be solved by radicals must be regarded as exceptional. The theory of groups gives the means of determining whether an equation comes under this exceptional case, and of solving the equation when it does. When it does not, the theory provides the means of reducing the problem presented by the equation to a normal form. From this point of view the theory of equations of the fifth degree has been exhaustively treated, and the problems presented by certain equations of the sixth and seventh degrees have actually been reduced to normal form.