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

48

SUBSTITUTION GROUPS

[CH. II

these two infinite systems are composed of all the possible
primitive groups which are of class 2 and class 3 respectively.
A substitution which actually contains exactly k letters is sometimes said to be of class k.
To prove that a primitive group G which is of class 2 and
of degree n is symmetric, it is only necessary to observe that
such a group contains at least two transpositions having a letter
in common, otherwise G would be imprimitive. Hence G
involves the symmetric group of degree 3. It must therefore
contain two such symmetric groups which have two letters
in common. Hence G contains the symmetric group of degree
4. By continuing this process, it results that G is the symmetric
group of degree n. In exactly the same manner, it can be proved
that if a primitive group of degree n contains a substitution of
the form abc it includes the alternating group of degree n. That
is, if a primitive substitution group involves a transposition it is
a symmetric group, and if it involves a substitution of the form abc
without also involving a transposition, it is an alternating group.
Suppose that G is a primitive group which contains a substitution of degree and of order p, p>3, and that G is of
degree n, n>p, p being a prime number. There must be at
least two substitutions in G which are of degree and of order p,
and which have some but not all their letters in common. Let
sl and S2 be two such substitutions. If S2 involves more than
one letter which is not also contained in si, there is some
power of Si in which two such letters are adjacent. The
transform of Si by this power will then be a substitution which
has more letters in common with sl than S2 has; but this transform involves at least one letter which is not contained in sl.
Hence we may assume that G contains two substitutions of
degree p and of order p such that these substitutions contain
exactly p-1 common letters. These two substitutions generate
a doubly transitive group of degree p+ 1. By continuing these
considerations it results that if a primitive group of degree n
contains a substitution of degree p and of order p, p being any
prime number, this primitive group is at least (n-p+1)-fold
transitive,