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

276

GROUP CHARACTERISTICS

[CHr. XIII

30. Going back to 10, it remains for us to prove that at least
one of the numbers f21, f31,.'., fkl is zero. To accomplish this we start with the following equations (which we shall
establish in 40):
2 2g-g'
1212+1312+... +fk1l2 2
- g'
(5) 121122+131132+ ~ 2 +fklfk 2 9 2
g
f222+f322+... +12 =g 1222+1,
9/2
from which we derive
(f22 - n (f1)2+(f 32 + n12~11)2+... (fk2-n'2fk1)2=1.
Now, since the terms in the left-hand member are positive
integers or zero, it follows that one of them is unity and the
rest zero; say
f22=n12f21k1, f32= fl2f31,. k.,fk2=nf2fkl.
Substituting in the second of the equations (5) and subtracting fl'2 times the first gives us = 0,21 =0 what we set out
to prove.
40. To prove (5), consider the sum
M= Xv1X1161t+Xv1X2102t+... +Xf1Xg'l0g't
+XV2X12611t+ Xv2X2202t+ +Xv2Xg'20g't
+XvkX1kOllt+XvkX2k1O2t+.. +XkXg'kOg't (1 < v g').
Let there be I' operators conjugate to T, in G', and there will
be I'glg' conjugates to T. in 'G. Then, adding by columns,
we get (~ 135, Ex. 2)

M =h'(Xvlxv +Xv2Xv2+... +XvkXvk)olt =9'6VI.