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

84

ABSTRACT GROUPS

[CH. III

invariant under G, H will correspond to a subgroup in a quotient
group, such that this quotient group can be represented transitively with respect to this subgroup. Hence the theorem
is also true in this case. For an abstract proof of this theorem
the reader may consult H. W. Chapman, Messenger of Mathematics, vol. 42 (1913), page 132.
EXERCISES
1. If a dicyclic group is represented as a transitive substitution group
it must be regular.
2. A dihedral group of order 2n, n>2, can be represented in two and
in only two ways as a transitive substitution group.
3. Only one of the five possible groups of order 8 can be represented
as a non-regular transitive substitution group.
4. If a simple group of composite order is represented as a transitive
group of lowest possible degree it must be primitive.
5. There are five and only five abstract groups of order 12.
34. Historical Note.* The concept of group is one of the
oldest mathematical concepts. Even in the development of
elementary geometry by the Greeks this concept played a
fundamental r6le, as was pointed out by H. Poincare in an
article entitled " On the foundations of Geometry," Monist,
volume 9 (1898), pages 1-43. It was, however, not developed
into an extensive theory until a comparatively recent period.
In the latter half of the eighteenth century various writers,
especially J. L. Lagrange and A. T. Vandermonde, began to
lay stress, in their algebraic investigations, upon the elements
of the theory of substitutions. On the other hand, L. Euler
brought some properties of abelian groups into prominence,
especially by his work on power residues. Towards the close
of the eighteenth and at the beginning of the nineteenth century, two Italian mathematicians, P. Ruffini and P. Abbati,
entered more directly on the study of substitution groups
by proving that there are no three or four-valued rational
functions of more than four variables, and that a two-valued
function must be alternating.
During the first half of the nineteenth century a number
of other investigators entered this field. Foremost among
* Cf. Bibliotheca Mathematica, vol. 10 (1910), p. 317.