is invariant for all equivalent representations, when written as a polynomial in [lambda]. Moreover, it has the same value for S and S', if these are two conjugate operations in G. Of the various invariants that thus arise the most important is s11 + s22 + ... + s_(mm), which is called the "characteristic" of S. If S is an operation of order p, its characteristic is the sum of m pth roots of unity; and in particular, if S is the identical operation its characteristic is m. If r is the number of sets of conjugate operations in G, there is, for each representation of G as an irreducible group, a set of r characteristics: X1, X2, ... X_r, one corresponding to each conjugate set; so that for the r irreducible representations just r such sets of characteristics arise. These are distinct, in the sense that if [Psi]1, [Psi]2, ..., [Psi]_r are the characteristics for a distinct representation from the above, then X_i and [Psi]_i are not equal for all values of the suffix i. It may be the case that the r characteristics for a given representation are all real. If this is so the representation is said to be self-inverse. In the contrary case there is always another representation, called the "inverse" representation, for which each characteristic is the conjugate imaginary of the corresponding one in the original representation. The characteristics are subject to certain remarkable relations. If h_p denotes the number of operations in the pth conjugate set, while X^_i{p}, and X^j{p} are the characteristics of the pth conjugate set in [Gamma]_i and [Gamma]_j, then __p=r \ h_p X_p^i X^_p^j = 0 or n, /__p=1 according to [Gamma]_i and [Gamma]_j are not or are inverse representations, n being the order of G. Again __i=r \ X_p^i X^_q^i = 0 or n/h_p /__i=1 according as the pth and qth conjugate sets are not or are inverse; the qth set being called the inverse of the pth if it consists of the inverses of the operations constituting the pth. Linear homogeneous groups. Another form in which every group of finite order can be represented is that known as a linear homogeneous group. If in the equations x'_r = a_(r1)x1 + a_(r2)x2 + ... +a_(rm)x_m, (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