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

~ 121

TRANSITIVE SUBSTITUTION GROUPS

31

represent two very important types of groups. In the former
each letter of the group is replaced by every other letter by
the various substitutions of the group. Such a group is said
to be transitive. In the latter of these two groups, there is a
letter which is not replaced by every other letter, and hence
this group is called intransitive. Every substitution group
evidently belongs to one and only one of these two types. Every
symmetric group and every alternating group is transitive.
Suppose that G is a transitive group on the n letters al,
a2,. ~ ~, an. There is at least one substitution in G which
does not involve the letter al, viz., the identity. In general,
the substitutions of G which omit a, constitute a subgroup
G1 of order gi. As G is transitive it must involve a substitution which replaces ai by a2, and this substitution transforms
G1 into G2, G2 being composed of all the substitutions of G which
omit a2. Hence G contains n conjugate subgroups Gi, G2,... G, each of which is composed of all the substitutions
of G which omit a letter. It is not necessary that all of these
n subgroups be distinct. In fact, in the octic group they
form two pairs of identical subgroups.
All the substitutions of G can be arranged in a rectangle
as follows, the substitutions of G1 forming the first row:
1) S2, S3 ~.., Sg$
t2 S2t2, s3t 2,., Sgt2
t3, S2t3, S3t3, e o Sg,t
tx, S2tx, S3tX.., sgltx
where X=g/gl. If t2 replaces al by a2 then all the gi substitutions of the row involving t2 have this property. If any other
substitution of G should replace ai by a2, all the distinct products, obtained by multiplying its inverse into itself and into
all the substitutions of the row involving t2, would transform
a2 into itself. As this would give more than gi distinct sub-stitutions, the row which involves t2 contains all the substitutions of G which replace al by a2.