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

PRODUCT OF CHARACTERISTICS

267

Accordingly, this product is the sum of one or more of the
matrices M1,..., Mn, possibly repeated a certain number
of times. Hence the lemma.
135. Theorem 20. Let the number of transformations in
the different conjugate sets of a transitive group G in n variables
be gi, g2,..., gh, and let the corresponding characteristics be
denoted by xl, X2,..., Xh (cf. ~ 89). Then
(1) ns)(n) C( nV) (s, t1, 2,...,
where Cstl,... represent certain positive integers or zero.
Proof. We substitute in the equation of Lemma 2 the
canonical forms of the matrices M1,..., Ma as given by
Lemma 1, and obtain the equation (,., ) =(7,... ),
where 3 has for value the left-hand member of (1), and y the
right-hand member.
EXERCISES
1. Selecting the h equations (1) obtained by keeping s fixed while taking
t= 1, 2,..., h, prove that gsxs/n is an algebraic integer (cf. ~ 116, 7~).
2. Prove that if Ss and St- are conjugate, then
k k
Xs) Xt ( = Xs Xs(i -
j=l j1=l
where the summation extends over a set of non-equivalent groups into
which the regular substitution group H breaks up; if S, and St - are not
conjugate, the first sum vanishes.
(Prove that = 'njxv(j) = 0 if Sv is not the identity; and that if it is, the
sum equals g. Prove also that if Ss and St 1 are conjugate, gs= gt, and in
the right-hand member of (1) we shall then have cstl=gs. We assume
nj= xi() to be the characteristic of the identity.)
COROLLARY 1. The quantity gsxs equals the product of n by
the sum of a finite number of roots of unity.
This follows from the statement of Exercise (1) and ~116,
7~
COROLLARY 2. The number of variables n of a transitive
linear group G is a factor of the order g.
Proof. The equation from Theorem 18 may be written
(2) g=glxixl+g2X2X2+ -.. +ghXhXh