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

172

ISOMORPHISMS

[CH. VII

uct of its Sylow subgroups, the I of an abelian group is always
the direct product of the I's of its Sylow subgroups.
67. Group of Isomorphisms of a Transitive Substitution
Group. Suppose that G is a transitive substitution group of
degree n which involves no subgroups of index n and degree n,
but involves subgroups of degree n-1. Its n subgroups of
degree n - 1 must therefore correspond among themselves
in every automorphism of G, and these subgroups- may be so
lettered that they are transformed by every substitution in G
in exactly the same manner as the letters of this substitution
are transformed. From this it results that if each of the n
subgroups corresponds to itself in any automorphism of G, each
of the substitutions of G must also correspond to itself in this
automorphism. That is, the I of G may be represented on
letters corresponding to these subgroups.
As G involves subgroups of degree n- 1, it is simply isomorphic
with its group of inner isomorphisms. Hence the I of G may be
represented as a transitive substitution group of degree n which
contains G invariantly. This proves the following theorem:
If G is a transitive substitution group of degree n which involves
subgroups of degree n-1 but no subgroups of both degree n and
index n, then the group of isomorphisms of G can be represented
as a transitive substitution group of degree n which contains G
as an invariant subgroup.
As the symmetric group of degree n involves no subgroup
of degree and index n when n 6, and as it contains a subgroup
of degree n-1 whenever n 2, it results from the given theorem
that the I of every symmetric group of degree n, except when
n is either 2 or 6, can be represented as a substitution group
on n letters, which contains this symmetric group. This substitution group must therefore be the corresponding symmetric group, as was proved above. In a similar way we may
observe by means of this theorem that the metacyclic group
of degree p and of order p(p -1) is its own group of isomorphisms. These illustrations may suffice to point out the usefulness of this theorem in the study of the groups of isomorphisms
of substitution groups.