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. Groups of linear substitutions. 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 to each operation S of a group G of finite order there will correspond a linear substitution s, viz. __j=m x_i = \ s_(ij)x_j(i, j = 1, 2, ..., m), /__j=1 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.