SUBSTITUTION (Lat. substitutio, a putting in the place of another, from substituerc, to put in the place of another, from sub, under + statucrc. to place, from stare, to stand). A mathematical operation by which one expression is replaced by another. The term has, however, conic to have a technical meaning in modern mathematics, and this has led to an important branch known as the theory of substitutions. If a elements, a,. ... a. are given, and a. and a,, a„, a,, ... a. are two ar rangements of these elements, the operation of passing fmm the first of these arrangements to the second is called a substitution of the 71 ele ments. It follows that there are no substitutions of a elements, including the identical substitu tion, which leaves the order of the letters un changed. A substitution which in place of the arrangement a,„ a,. a,, a„ gives a'„ a'„ a'„ is represented by the symbol If, however, a, is replaced by a,, a, by a,, an-i by a„, and a„ by the substitution is said to he cyclic, and is more conveniently repre sented by ...a „), or even by (I, 2, 3, n), than by the more elaborate symbol Ca; Similarly a substitution like may be written (oed) (bef), meaning that while a chances to c, c to (1, and d to «, b at the same time changes to c, c to f. and f to b. This sym bolism is further extended thus: Consider (ab) (ae): this means that a changes to b, b to a, and a to c. and c back to a. a result which evi dently may also be indicated by (abc), so that (ab) (ac) = (abc). But the Caine reasoning shows that (ac) (ab) = (acb). Hence if s, = (ab) and s, = (ac), s,s, 4- s,s,. For conveni ence. s,s, is called the product of and s, in the order given, from which it appears that the commutative law of multiplication does not hold true in the theory of substitution. If in the product s,s,s, s. we have s, = 8„, the product is called the power of each substitution. If a substitution leaves all the elements un changed in order it is called an identical substi tution and is represented by 1. If the product of two substitutions, like and is 1, each is called the inrcrsc of the other, and if the first is represented by s, the second is represented by equaling I.
A collection of substitutions is said to form a group, if the product of any two is another of the same collection. This may be illustrated outside the field of substitutions by the three cube roots of unity 1,— ?; ?ei A the product of any two being another of the same collection. The six substitutions = I, (.ryz), (.1.,:y), x(yz), s,= y(.7.r),
(.8y) also form a group. The number of substitutions of a group is its order, and this is always a factor of n!. Thus in the group given the order is 6, and this is a factor of 3!. if all substitutions of a group H are contained in another group G, H is called a sub-group of G and the order of II is a factor of that of G. A group whose operations are all permutable with one another is called an Abelian group.
Lagrange (1770) was one of the first to under take a scientific treatment of substitutions in connection with the theory of the quintic equa tion. He invented a 'calcul des combinaisons,' the first real step toward the theory of substi tutions, P.nfiini (1799) was the next to under take a serious study of the subject. again in the attempt to show the impossibility of solving the quintie. To Galois (q.v.), however, the honor of establishing the theory is usually ascribed.
He found that if r„ r„, r„ r. are the a roots of an equation, there is always a group of per mutations of the r's such that (1) every func tion of the roots invariable by the substitutions of the group is rationally known, and (2), re ciprocally, every rationally determinable func tion of the roots is invariable by the substitu tions of the group, a discovery that eventually led to the proof of the insolubility of the quintie (Liouville's Journal, vol. xi.). Consult also Wurrcs mathematiques de Galois (Paris, 1897). Cauchy was the first of the well known French mathematicians to recognize the importance of the theory, and numerous im portant propositions are due to him (Journal de l'ecole polytechnique. 1815; Excreices d'analysc at de physique nmtbwmatique, vol. iii., Paris, 1844). Serret was the first to give a connected account of the theory ((lours cralgrbre superi cure. in the 3d ed., Paris, 1866). This was followed by Jordan's Trait(' des substitutions et des equations algebriqucs (Paris, 1870). Sylow (1872) was the first to treat the subject apart from its applications to equations (Math. An 'mien, vol. v.). He was followed by Nctto, whose Substitzttionsthcoric (Leipzig, 1882) was translated by Cole (Ann Arbor. 1892), thus making the theory accessible to English readers. Burnside, Theory of Groups of Finite Order (Cambridge. 1897). has brought the theory even more prominently before English and American scholars. The first to attempt to simplify the applications of the theory to the subject of equa tions in an elementary text-book was Petersen, Thcoric der algebr«ischen Glcichungcn (Copen hagen, 187S).