x′i = Σsij xj (i, j = 1, 2, ..., m),

is the linear substitution which, in a given irreducible representation of a group of finite order G, corresponds to the operation S, the determinant

s11 − λs12...s1m
s21s22 − λ...s2m
......
......
......
sm1s2m...smm − λ

is invariant for all equivalent representations, when written as a polynomial in λ. 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 + ... + smm, 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, ... Xr, 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 Ψ1, Ψ2, ..., Ψr are the characteristics for a distinct representation from the above, then Xi and Ψi 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 hp denotes the number of operations in the pth conjugate set, while Xip, and Xjp are the characteristics of the pth conjugate set in Γi and Γj, then

Σp=rp=1 hp Xip Xjp = 0 or n,

according to Γi and Γj are not or are inverse representations, n being the order of G.

Again

Σi=ri=1 Xip Xiq = 0 or n/hp

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.

Another form in which every group of finite order can be represented Linear homogeneous groups. is that known as a linear homogeneous group. If in the equations