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

~ 33] ABSTRACT GROUP REPRESENTED BY SUBSTITUTIONS 83

morphisms of G, and such that these subgroups do not include
any invariant subgroup of G besides the identity.*
For instance, the symmetric group of order 24 contains three
conjugate cyclic subgroups of order 4 and also three conjugate
noncyclic subgroups of this order. It contains no other noninvariant subgroup of order 4. Hence this symmetric group
appears twice among the transitive substitution groups of
degree 6. As the alternating group of degree 4 contains two
sets of conjugate subgroups, of orders 2 and 3 respectively,
this group can be represented transitively in only two ways
besides the regular form. That is, it appears as a transitive
group of degree 4 and also as a transitive group of degree 6.
It should be observed that these considerations establish another
very close contact between the theory of abstract groups and
that of transitive substitution groups.
It is now easy to prove the theorem, to which we referred
in ~ 10, that the co-set multipliers may be so selected that they
are the same on the right as on the left. When the subgroup
H is invariant this requires no proof. When H does not involve
any invariant subgroup of G besides the identity, and G is
represented as a transitive substitution group of degree n with
respect to H, it may be supposed that H is composed of all the
substitutions of G which omit a given letter a. The right
co-sets will then be composed separately of all the substitutions
of G in which a is replaced by a given letter. In the left co-sets
a is replaced by all the letters of a transitive constituent of H
if H is intransitive on n-1 letters. If H is transitive on n-1
letters the theorem is evident.
Hence it remains only to consider the case when H is intransitive. In this case it is evidently possible to select a certain
number of left co-sets involving all the substitutions of the
same number of right co-sets, and in these the multipliers
may be made the same in the left co-sets as in the corresponding
right co-sets. Hence the theorem is established in case H contains no invariant subgroup of G besides the identity.
If H involves such an invariant subgroup but is not itself
* Bulletin of the American Mathematical Society, vol. 3 (1897), p. 215.