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

46

SUBSTITUTION GROUPS

[CH. II

19. Holomorph of a Regular Group. If G is a regular group
of order n, all the substitutions on these n letters which transform G into itself constitute a group which has been called the
holomorph * of G. For instance, the symmetric group of
degree 3 is the holomorph of its subgroup of order 3, (abcd)s
is the holomorph of (abcd), and (abcde)20 is the holomorph of
(abcde). The holomorph of G includes the conjoint of G, and
hence it is also the holomorph of this conjoint. If this conjoint
is not identical with G, it is conjugate with G under a substitution of order 2 which transforms the holomorph of G into itself.
This substitution and G generate a group known as the double
holomorph of G. Every non-abelian group has a double
holomorph.
Since the holomorph K of G involves exactly n substitutions
which are commutative with every substitution of G, it must
transform the substitutions of G in k/n different ways, k being
the order of K. The largest subgroup of degree n-1 contained in K must also transform the substitutions of G in k/n
different ways. This subgroup is known as the group of
isomorphisms of G.t Hence the order of the holomorph of a
group is the product of the order of the group and the order of its
group of isomorphisms. Since any two simply isomorphic
regular groups are conjugate, the group of isomorphisms of G
transforms G into every possible simple isomorphism with itself.
That is, it transforms it into all its possible automorphisms.
If a group involves substitutions which transform it into
every possible automorphism, but does not contain any invariant substitution besides the identity, it is said to be a complete
group. The symmetric group of degree 3 is evidently a complete group. The holomorph of a complete group is the product
of the group and its conjoint. It is easy to prove that the
* The concept of holomorph was used by many early writers, but the term
was introduced by W. Burnside in the first edition of his Theory of Groups,
1897, p. 228.
t The statement relating to this matter in the Encyklopddie der Mathematischen
Wissenschaften, vol. 1, p. 221, note 103, is inaccurate. The group of isomorphisms
is one of the most important and also one of the most far-reaching concepts in
group theory.