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

40

SUBSTITUTION GROUPS

[CH. II

to which L belongs is transformed according to an imprimitive
group there must be such a group as H which includes K, since
all the substitutions which transform among themselves the
substitutions or subgroups, corresponding to a system of imprimitivity, must constitute a group. Hence the given general
theorem is established.
To deduce the special case relating to the primitivity of
G when G1 is of degree n-1, we have only to observe that a
necessary and sufficient condition that G1 is transformed into
itself by only its own substitutions, is that its degree is n-1.
As G transforms the conjugates of G1 in exactly the same way
as it transforms its letters, whenever the degree of G1 is n-1,
and as we assume in the present case that the degree of G1 is
exactly n-1, it results directly from the given theorem that G
transforms the conjugates of G1, and hence also its letters,
according to a primitive group whenever G1 is maximal, and
only then.
If Gi is a transitive group on n-1 letters, G is said to be
doubly transitive, and every pair of letters of G is transformed
into every other pair by the substitutions of G. In general,
G is said to be r-fold transitive, whenever G1 is (r - 1)-fold transitive and of degree n-1. Since the order of a transitive group
is always a multiple of its degree, it results that the order of
an r-fold transitive group is a multiple of n(n-1).
(n-r+1). A group which is more than simply transitive
is said to be multiply transitive. The alternating group of
degree n is (n-2)-fold transitive and the symmetric group of
degree n is said to be either n-fold or (n-l)-fold transitive. We
shall generally say that this group is (n-1)-fold transitive.
With the exception of the alternating and the symmetric groups
no group is known which is more than five-fold transitive and
only two such five-fold transitive groups are known. These
groups are of degrees 12 and 24 respectively and were discovered by E. Mathieu in 1861. The theory of multiply
transitive groups has not yet been extensively developed, and
it seems to offer great difficulties.