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

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COMMUTATIVE SUBSTITUTIONS

19

the two substitutions ab cd, ac bd are commutative, but ab cd
is not commutative with be. Two substitutions which have
no letter in common are always commutative.
It is often necessary to find all the substitutions on certain
letters which are commutative with a given substitution. The
solution of this problem is based on finding all the substitutions
on the letters al, a2,..., an which are commutative with
the cyclic substitution sl=aa2... an. It is clear that si
is commutative with all of its powers, and hence sl is commutative with at least n substitutions, including the identity,
on the letters ai, a2,..., an.
All substitutions which are commutative with si must also
be commutative with s",al where ai is any positive integer.
Suppose that t2 is a substitution on the letters al, a2,..., a
which is commutative with si but is not a power of si. It is
evident that t2 must involve each of the letters al, a2,..., an.
Hence we may suppose that t2=aa,..., where a is one
of the numbers 2, 3,..., n. Since sl-l=aa,..., it
results that t2sln+l-a is a substitution which is commutative
with si, does not involve al, and is not the identity. That is,
we arrive at an absurdity by assuming that more than n substitutions on the letters ai, a2,..., a, are commutative with
sl. This proves the theorem: The only substitutions on n letters
which are commutative with a cyclic substitution on these letters
are the powers of this cyclic substitution.
If a substitution s2 is composed of X cycles such that no
two of these cycles involve the same number of letters, then
all the substitutions on the letters of 52, which are commutative
with S2 must also be commutative with each cycle of s2. The
number of the substitutions which are commutative with s2,
and involve only letters contained in s2, is therefore equal to
the product of the orders of the cycles of s2. For instance,
the substitutions which are commutative with the following
substitution
abcde.fgh,
and involve only its eight letters, constitute a substitution
group of order 15.