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

50

SUBSTITUTION GROUPS

[CH. II

The fact that this primitive group actually exists was proved
by C. Jordan,* but we shall not give the proof here. We have,
however, established the following theorem:
When the prime number p, p>3, is not of the form 2n-1
there is one and only one primitive group of class p. When p is
of the form 2"-1 there cannot be more than three primitive groups
of class p.
The study of primitive substitution groups by means of
their classes was begun by C. Jordan, who proved that there is
a finite number of such groups of every class. In recent years
W. A. Manning has contributed new theorems on this interesting but difficult subject.t
EXERCISES
1. If a primitive substitution group contains two transitive subgroups
which can be transformed into each other by a transposition, the primitive
group is alternating or symmetric.
2. Prove that the two substitutions abe cdf and ag. bf generate a group
of order 168, and that this group is simple.
3. It is known that the simple group of order 504 can be represented
as a transitive substitution group on 9 letters. Hence prove that it contains the abelian group of order 8 which is composed of seven substitutions of order 2 and the identity.
* Journal de Mathematiques (2), vol. 17 (1872), p. 351.
t W. A. Manning, Transactions of the American Mathematical Society, vol.
11 (1912), p. 375.