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

234

SOME SPECIAL TYPES OF GROUPS

[CH. XI

letters of I form a transitive set for P; accordingly, there
is a substitution in P, say Si, which also changes yi into y2.
Then RS~1- leaves yi fixed and must therefore be the identity;
that is, R=S1.
It follows that the substitutions of G transform I into
just g/pX functions I, II, 12,..., no one of which is a constant multiple of another. The product K=11112.... of
these functions is therefore an invariant of G.
Now, all the substitutions of G for which K is an absolute
invariant, that is, for which (K)R=K, must form an invariant
subgroup G' of G, as is easily seen. To this group the substitution T does not belong, since
(K) T = K,
where k = g/pX. For, let I = (I)R, then
(Ir)T = (I)RT = (I) TR = 0(I)R = 0Ir.
But, kZ 1 since k is prime to p.
This subgroup G' is of index p. For, the constant multipliers of K that result by operating upon K by the various
substitutions of G are integral functions of 0 and must be
roots of unity. Such roots can therefore, by ~ 116, 6~, be no
others than powers of 0. Moreover, it is readily seen that
each power must occur equally often, so that the power 1
occurs g/p times. Hence the theorem.
EXERCISES
1. Prove that a linear group in 3 variables of order 9g in which there is
no transformation of order 94 must contain similarity-transformations.
2. Prove that a linear group in n variables which contains a subgroup
F of similarity-transformations of order pk (p a prime > n), contains an
invariant subgroup of index pt, to which F does not belong.