An introduction to the modern theory of equations, by Florian Cajori.

ON SUBSTITUTIONS

105

Ex. 1. Show that (xyzw) is the same substitution as (wxyz).
Ex. 2. Show that (ala2 *.. an) is equal to
(aC^nC —T.?n1+ - - - a^ l2 *- aIn —.-l);
that, therefore, the same substitution may be represented in several ways
and that its form is consequently not unique.
82. Product of Substitutions. By the notation (acla.. ct,),
(blb2... b,) we mean that the substitution (aca.. "a,) is performed first; then, upon the result thus obtained, the substitution
(bi b2.. b,)
is performed. We call the two substitutions, placed in juxtaposition, their product in the given sequence.
If the product (1 2 3)(4 5 3) be applied to the digits
1 2 3 4 5, taken in their natural order, the substitution
(1 2 3) yields the arrangement 2 3 1 4 5. The substitution
(4 5 3) applied to this result gives the arrangement 2 4 1 5 3.
But this last arrangement may be obtained from the first by
the substitution (1 2 4 5 3). Hence the product of (1 2 3)
and (4 5 3) is equivalent to the single substitution (1 2 4 5 3).
The indicated product (1 2 3) (4 5 3) may be carried out conveniently
as follows: 1 is replaced by 2 in the first substitution, and 2 is not replaced in the second substitution; hence 1 is replaced by 2 in the product. Again, 2 is replaced by 3 in the first substitution, 3 is replaced by
4 in the second substitution; hence 2 is replaced by 4 in the product.
Likewise, 4 is replaced by 5 in the second substitution and also in the
product; 5 is replaced by 3 in the second substitution and in the product.
Hence the result of the multiplication is the substitution (1 2 4 5 3).
Ex. 1. Show that (4 5 3)(1 2 3)=(1 2 3 4 5).
Ex. 2. Show that (abcd)(acde)=(abdce).
83. Commutative and Associative Law. Notice that the
product of (1 2 3)(4 5 3) is not the same as the product of
(4 5 3)(1 2 3). On the other hand, we see that (1 2 3)(4 5)
= (4 5)(1 2 3) and that (xy)(zt)(xz)(yvs) (xz)(yyw)(xy?)(zw?').