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

274

GROUP CHARACTERISTICS

[CH. XII]

139. Theorem 24. A transitive substitution group of degree
n and class n-1 contains an invariant subgroup of degree and
class n.*
Proof. 1~. Let g be the order of G, and g'=g/n the order
of that subgroup G' of G whose substitutions leave fixed a
given letter. Again, let H represent the regular substitution
group simply isomorphic with G, and H' the regular substitution group simply isomorphic with G'. The groups H and
H' are of degrees g and g' respectively.
Now let H be resolved into its component linear groups,
from which we select a set of non-equivalent groups Hi, H2,..., Hk, the number of whose variables are respectively
nl=l, n2,..., nk. Similarly, let HI be resolved into its
different linear groups, from which we select a set of nonequivalent groups H'1, H'2,.., H',, in respectively n' = 1,
n 2,..., n' variables. The latter set and their equivalent
groups are all irreducible components of that subgroup of H
which corresponds to H'; they are, in fact, contained as irreducible components in the subgroups of H1,..., H, (and
their equivalent groups) which correspond to G' of G. Let us
suppose, for any subscript s< k, that the subgroup of Hs which
corresponds to G' of G breaks up intofis groups equivalent to
H'1, fs2 groups equivalent to H'2, etc. This division may be
exhibited clearly by the following equation:
\Hs I=fslH' +fs2H'2+... +fslHIZ
Evidently, I H =H'1, so that
fll=l f12=0,...,fl=0.
Moreover, (~ 132, Ex. 3),
n12+n22+... +nk2=g, n'2+n'22+... +-n'2=g'.
Again, if T1(=the identity), T2,..., Tg, are the transformations of I Hs I, and Xis(=ns), X2s,.., Xg,' their char* Frobenius, Sitzungsberichte, etc., 1901, pp. 1223-1225.