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

~ 14] SUBSTITUTIONS INVARIANT UNDER A GROUP

37

The two groups G and G' which are defined by Jordan's
theorem are called conjoints. When G is abelian it coincides
with its conjoint and vice versa. A special case of this theorem
relating to a cyclic group was proved above in ~ 8. Suppose
that G is transitive and of degree n, but not regular, and that
all the substitutions of G which omit al omit also a2,,
a. Hence G1, which is composed of all the substitutions of G
which omit ai, is transformed into itself by all the substitutions
of G which replace ac by a2,..., a. That is, GI is transformed into itself by a subgroup H of order agi. All of the
substitutions of FH, except those of G1, must involve each of
the letters al, c2, o., a. Hence H is an intransitive group,
and the components of its substitutions involving the letters
al, 2,..,,a must form a regular group. This regular
group is a transitive constituent of H and the subgroup Gi,
which is transformed into itself by H, corresponds to the identity of this transitive constituent. Let Ci be the conjoint
of this constituent, and consider the n/a transforms of C1
under G.
A (1, 1) correspondence can be established between the
substitutions of these n/a transforms such that each substitution is of degree nz and is transformed into itself by G. To do
this it is only necessary to regard as one substitution all the
transforms of a single substitution of Ci. This proves a theorem
due to H. W. Kuhn,* which is a generalization of Jordan's
theorem and may be stated as follows:
A necessary and sufficient condition that there are a substitutions on the letters of a transitive group, which are commutative with every substitution of the group, is that its subgroup which
is composed of all its substitutions which omit one letter omits
exactly a letters /
If GC is of degree nz-1, the identity is therefore the only
substitution on these n letters which is commutative with
every substitution of G, as is also otherwise evident.
* American Journal of Miathcmatics, vol. 26 (1904), p. 67.