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

~ 201

CLASS OF A SUBSTITUTION GROUP

49

It is now easy to prove that a primitive group of class p,
p>3, cannot have a degree which exceeds p+2. In fact,
if such a group were of degree p+3, it would be at least fourfold, or four times, transitive. Since any set of four letters
of such a group can-be replaced by an arbitrary set of four letters, it results that any four times transitive group must involve
an intransitive subgroup H which has the symmetric group of
degree 3 for one constituent, and a transitive group on the
remaining letters for the other constituent. In the present
case the latter group is of degree p.
In any transitive group of degree p, the subgroups of order
p generate a simple group, since an invariant subgroup of a
primitive group is transitive. If this simple group is not the
entire group, it must be invariant under the entire group and
the corresponding quotient group must be cyclic, since it is a
subgroup of the group of isomorphisms of a group of order p,
and this group of isomorphisms is cyclic, since p has primitive
roots. Hence it results that H includes substitutions of the
form abc whenever G is of degree p+3, since its transitive constituent of degree p cannot give rise to a quotient group which
is simply isomorphic with the symmetric group of degree 3
when p>3. That is, if a primitive group is of class p, p being
a prime number greater than 3, the degree of this primitive group
is at. most p+2.
There is evidently one and only one primitive group of
degree p and of class p; viz., the group of order p. In order
that a primitive group of degree p+1 be of class p, it is clearly
necessary that this group be of order p(p+l), and that it
contain (p+l)(p-1) substitutions of order p. Hence p+1
must be of the form 2m, and if p+1 is of this form there is one
and only one such group. A primitive group of degree p+2
which is of class p must therefore include this primitive group
of degree p+l. It must also contain a substitution of order
2 and of degree p+1 which transforms into its inverse one of
the substitutions of order p in this group of degree p+ 1. Hence
there cannot be more than one primitive group of degree
p+2 and of class p.