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

ON SUBSTITUTIONS

109

88. Theorem. The product t-1st may be conveniently obtained
from the substitutions s and t by performing upon each cycle of s
the substitution t.
Let s = (abc... )'b'c' *. )*..
and t = (abc... a'b' c'.)
Take any one of the letters a, fi, y, ', a(', /', y', *, say P.
By t-1, / is replaced by b; by s, b is replaced by c; by t, c is,
replaced by y. Hence by t-lst, f3 is replaced by y.
Now, if by t we substitute / for b and y for c in the cycles
of s, then, instead of the sequence b c, we have in s the sequence
/ y, which replaces f3 by y, as before. As this consideration
applies not to / alone, but to any letter, the theorem is
established.
In the operation t-lst, t is said to transfonrm s; the operation
is called a trancsformation.
Ex. 1. If s= (12 3)(4 56 7), t = (57 2 3), then t- (2 7 5). To
illustrate the theorem just proved, apply t-1 to the arrangement 1 2 3 4 5 6 7
and we get 1 7 2 4 3 6 5. To this result apply the substitution s, and we
have 2 4 3 5 1 7 6. To this last arrangement apply t, and we obtain
finally3 4 5 7 2 6.
This same final arrangement is obtained more easily, if in place of performling the three substitutions, we perform upon the arraingement
1 2 3 4 5 6 7 only one substitution, namely s' = (1 3 5) (4 7 6 2). Now s' is
gotten from s by performing upon each cycle of s the substitution t.
Ex. 2. If s = (1 2 3)(4 5 6 7) and t - (2 4 3 7), find t-st by theorem
in ~ 88.
Ex. 3. If s (ab)(cd), t — (abc), determine the result of operating
with t-.st upon the arrangement a b c d.
89. Transpositions. A transposition is a cyclic substitutin containing two elements. Thus, (ab), (bc), (1 2) are
t rallsl)ositions.
Ex. 1. sh,w that the square of any transposition is the identical
slubstiutitni,.c.!.