It has been seen at the beginning of this section that every group of finite order N can be presented as a group of permutations (i.e. linear substitutions in a limited sense) on N symbols. This group is obviously reducible; in fact, the sum of the symbols remain unaltered by every substitution of the group. The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following. If r is the number of different sets of conjugate operations in the group, then, when the group of N permutations is completely reduced, (i.) just r distinct irreducible representations occur: (ii.) each of these occurs a number of times equal to the number of symbols on which it operates: (iii.) these irreducible representations exhaust all the distinct irreducible representations of the group. Among these representations what is called the "identical" representation necessarily occurs, i.e. that in which each operation of the group corresponds to leaving a single symbol unchanged. If these representations are denoted by [Gamma]1, [Gamma]2, ..., [Gamma]_r, then any representation of the group as a group of linear substitutions, or in particular as a group of permutations, may be uniquely represented by a symbol [Sigma][alpha]_i[Gamma]_i, in the sense that the representation when completely reduced will contain the representation [Gamma]_i just [alpha]_i times for each suffix i. Group characteristics. A representation of a group of finite order as an irreducible group of linear substitutions may be presented in an infinite number of equivalent forms. If x'_i = [Sigma] s_(ij)x_j (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 - [lambda] s12 ... s_(1m) | | s21 s22-[lambda] ... s_(2m) | | . . ... . | | . . ... . | | . . ... . | | s_m1 s_2m ... s_(mm) - [lambda] |