x′r = ar1x1 + ar2x2 + ... + armxm, (r = 1, 2, ..., m),
which define a linear homogeneous substitution, the coefficients are integers, and if the equations are replaced by congruences to a finite modulus n, the system of congruences will give a definite operation, provided that the determinant of the coefficients is relatively prime to n. The product of two such operations is another operation of the same kind; and the total number of distinct operations is finite, since there is only a limited number of choices for the coefficients. The totality of these operations, therefore, constitutes a group of finite order; and such a group is known as a linear homogeneous group. If n is a prime the order of the group is
(nm − 1) (nm − n) ... (nm − nm−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
| z′ ≡ | az + b | , ad − bc ≠ 0, (mod. p) |
| cz + d |
constitutes a group of order p(p2 − 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 ≡ ar1x1 + ar2x2 + ... + armxm, (r = 1, 2, ..., m),