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

~1391 cGRwuIPS OF DEGREE n AND CLASS n-i 277
Again, substituting for _Xt from (3) and adding by lines we
bhave (~ 131, Cor. 1):
Al _(XVifi + Xv2f2t +..+ Xvkf (1t9)1t+~2t62z+..~Og'10g'I)
- (frifrt+ Ov2fr2frt +...+Ov~frifrt)g'.
r=1
Hence, equating the two values of M obtained we get
(6) 0= fr lfrt+ 0v2 ~fr2frt + + Ov(1 frt)
+... + OV fi~f~rt A I+Ov2A 2+.. + OvIA,
say, where the summation extends from r = i to r =k, and 1K<v ~g b'.
We can now prove the following equation:
where
g-g
q = 2
g/
When v> 1 the equation reduces to the following bfmeans of (6):
0= -q(1 +n'26Jv2 +... Ov
which is true, since the quantity in the parenthesis is the characteristic corresponding to T, in the regular group H' and is
therefore zero. If v = 1, the right-hand member of (7) become's
(cf. 1 0)
A1 -q+n' 2(A2- qn'2) +. +n',(A - qn I1)
=Ai~n'2A2+...+n'jAz-q(1+n1'2 2+. nZ2
= frj~frj+nl2fr2+...+n/z fri) -n't9- qg'
(8) Zfrtnr - C
Now, r4i1ftfr is the number. of times the group H', (or
equivalent groups) enters as a component of the subgroup H
of H. In this group, G' will evidently be represented as an