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

164

ISOMOIRPHI SMS

[CH. VII

corresponding to the operators of G, then G is either the abelian
group of order 2m and of type (1, 1, 1,... ) or G transforms
transitively the set of operators to which the letters of I correspond. As every multiply transitive group is primitive, this
theorem applies to all multiply transitive groups as well as to
the simply transitive primitive groups.
EXERCISES
1. If G is the symmetric group of degree 4 its I may be represented as
an imprimitive group of degree 6 on letters corresponding either to its
operators of order 4, or to its six conjugate operators of order 2. These
two imprimitive groups are not conjugate as substitution groups, since
the one is composed of positive substitutions, while the other contains
negative substitutions.
2. The group of isomorphisms of the symmetric group of degree n,
n#6, can be represented as a transitive substitution group on the n(n- 1)/2
letters corresponding to the transpositions of the symmetric group. When
I is thus represented, it is a simply transitive primitive group whenever
n>4.
Szggestions: Use the theorem that the symmetric group of degree
n, nf6 and n>2, is its own group of isomorphisms, and that it has no
outer isomorphisms. See ~ 65.
3. Prove that if the I of the quaternion group is represented as a substitution group whose letters correspond to its operators of order 4, it will
be conjugate with the group in the first of these exercises which involves
negative substitutions.
64. Doubly Transitive Substitution Groups of Isomorphisms.
If I is doubly transitive on letters corresponding to operators
of G, each of these operators generates a cyclic subgroup (s)
which is transformed into itself under the holomorph of G
by a subgroup composed entirely of operators which are commutative with s; for, if a complete set of conjugate operators
of G under I includes at least two powers of the same operator,
the operators of this system must be transformed according
to an imprimitive group. Suppose that si and s2 are two operators of G which correspond to letters of I. We may assume
that si, S2 are non-commutative; for, if all such operators were
commutative, G would be abelian and hence the order of every
operator of G would divide 2. Since this case is so elementary,