group or the direct product of a number of simply isomorphic
simple groups.
It has been seen that every group of finite order can be represented as a group of permutations performed on a set of symbols whose number is equal to the order of the group. In general such a representation is possible with a smaller number of symbols. Let H be a subgroup of G, and let the operations Permutation-groups. of G be divided, in respect of H, into the sets H, S2H, S3H, . . ., SmH. If S is any operation of G, the sets SH, SS2H, SS3H, . . ., SSmH differ from the previous sets only in the sequence in which they occur. In fact, if SSp belong to the set SqH, then since H is a group, the set SSpH is identical with the set SqH. Hence, to each operation S of the group will correspond a permutation performed on the symbols of the m sets, and to the product of two operations corresponds the product of the two analogous permutations. The set of permutations, therefore, forms a group isomorphic with the given group. Moreover, the isomorphism is simple unless for one or more operations, other than identity, the sets all remain unaltered. This can only be the case for S, when every operation conjugate to S belongs to H. In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.
The fact that every group of finite order can be represented, generally in several ways, as a group of permutations, gives special importance to such groups. The number of symbols involved in such a representation is called the degree of the group. In accordance with the general definitions already given, a permutation-group is called transitive or intransitive according as it does or does not contain permutations changing any one of the symbols into any other. It is called imprimitive or primitive according as the symbols can or cannot be arranged in sets, such that every permutation of the group changes the symbols of any one set either among themselves or into the symbols of another set. When a group is imprimitive the number of symbols in each set must clearly be the same.
The total number of permutations that can be performed on n symbols is n!, and these necessarily constitute a group. It is known as the symmetric group of degree n, the only rational functions of the symbols which are unaltered by all possible permutations being the symmetric functions. When any permutation is carried out on the product of the n(n − 1)/2, differences of the n symbols, it must either remain unaltered or its sign must be changed. Those permutations which leave the product unaltered constitute a group of order n!/2, which is called the alternating group of degree n; it is a self-conjugate subgroup of the symmetric group. Except when n = 4 the alternating group is a simple group. A group of degree n, which is not contained in the alternating group, must necessarily have a self-conjugate subgroup of index 2, consisting of those of its permutations which belong to the alternating group.
Among the various concrete forms in which a group of finite order can be presented the most important is that of a group of linear substitutions. Such groups have already been referred to in connexion with discontinuous groups. Here the number of distinct substitutions is necessarily finite; and Groups of linear substitutions. to each operation S of a group G of finite order there will correspond a linear substitution s, viz.
on a set of m variables, such that if ST = U, then st = u. The linear substitutions s, t, u, . . . then constitute a group g with which G is isomorphic; and whether the isomorphism is simple or multiple g is said to give a “representation” of G as a group of linear substitutions. If all the substitutions of g are transformed by the same substitution on the m variables, the (in general) new group of linear substitutions so constituted is said to be “equivalent” with g as a representation of G; and two representations are called “non-equivalent,” or “distinct,” when one is not capable of being transformed into the other.
A group of linear substitutions on m variables is said to be “reducible” when it is possible to choose m′ (< m) linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called “irreducible.” It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.
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 Γ1, Γ2, . . ., Γr, 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 ΣαiΓi, in the sense that the representation when completely reduced will contain the representation Γi just αi times for each suffix i.
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. Group characteristics.If
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 − λ | s12 | . . . | s1m |
| s21 | s22 − λ | . . . | s2m |
| . | . | . . . | . |
| . | . | . . . | . |
| . | . | . . . | . |
| sm1 | s2m | . . . | 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
according to Γi and Γj are not or are inverse representations, n being the order of G.
Again
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 is that known as a linear homogeneous group. If in the Linear homogeneous groups.equations
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 |
