Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

20

SUBSTITUTION GROUPS

[CH. Ii

A substitution s3 which involves k equal cycles is commutative with substitutions which permute these cycles according
to the symmetric group of degree k. If each of these cycles
involves n letters, the substitutions on these kn letters which
are commutative with S3 must therefore constitute a substitution group of order nk k!. For instance, the substitutions
on the nine letters involved in the following substitution
abc def ghi,
and which are commutative with this substitution, constitute
a group of order 27 6 = 162.
9. Transforms of a Substitution and of a Substitution Group.
If s and t represent any two substitutions, it is possible to
find a third substitution by means of the operation s-lts=tl,
where s-1 represents the inverse of s. The substitution t/
is called the transform * of t as regards s. There is a very
simple rule for deriving ti if s and t are given. We proceed to
develop this rule.
Suppose that
t=... aaa...
S=-... aaaat... a't...
It is not assumed that a, or a, actually appears in s when s is
written in the normal form, since a', may be identically equal
to a, and a', may be identically equal to aa. Hence the given
notation is entirely general. It is easy to see that
t=... a'a'...
That is, to obtain the transform of t as regards s we simply replace
each letter in t by the one by which s replaces this letter.t For
instance, if s=abcd.ef and t=ab'ce, then s-lts=bc.df. In
*This transformation is fundamental in the theory of groups. Many of
its properties were developed by E. Betti in 1852; Annali di Scienze matematiche
e fisiche, vol. 3,.p. 55.
t This simple method to find the transform of a substitution is found in C.
Jordan's thesis, Paris, 1860, p. 14.