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

108

ITHEORY OF EQUATIONS

87. Finite Number of Distinct Substitutions. The number of
distinct substitutions which can be performed upon a finite
number of elements aa,... a,, is finite, for the number of
substitutions cannot exceed the number of permutations, and
this is known to be finite. Hence, if upon ca,... a,, we perform an unlimited series of substitutions s, s2, s:, s4, *S, the
results of those substitutions cannot all be distinct. There
will be certain powers of s which give the same result as does s
itself. Let m -- 1 be the lowest power of this kind, then,s''+ = s.
This may be written s'"s -= s. Hence
s"nss-1 = ss1- = s = 1,
and s" = 1.
We call m the order of the substitution.
The order of a s1)/,stitution, is the least power of the substitution which is equivalentl to the identical substitution.
If S thenu2 ( 3 1234s
If s 34 ' 41 t 9 (12 2
4 2 3 4 2 3 4\ etc.
-1234/' 23 41/
Hence m + 1 = 5, m -= 4, and s4 = so = 1, sc = s2, and generally,
84n+r = Sr
This substitution s is cyclic. It is evident that the order o*f a
cyclic or circular substitution is equal to the nmlber of its eleaets
(digits).
If s- (1 2 3)(4 5), then s2= (1 3 2), s3= (4 5), s4= (12:',
5 = (1 3 2)(4 5), s = 1. Hence the order is 6.
If n1, n2, n3,... denote the number of elements in the successive
cycles of a substitution, then its order is a number exactly
divisible by each of the numbers n,, nt2, 3, *; that is, its order
is the least common multiple of n 1, ' n 3
Ex. 1. Show by actual substitution that the order of s = (1 2) (3 4 5).
(6 7 8 9) is 12 or the L. C. M. of 2, 3, 4.