259 
eee | 
units ete. We thus see, that A, contains only £k—Zy,y21 
principal units, from which follows y, = y, =...= y, = 1. Therefore 
the principal units may be arranged in such way 
oe 08 En 
that 
had EE | 
Bee Sen Phe woe ee a) 
L dil 
The same reasoning holding for the M the principal units may 
be arranged in such a way 
Deere eed „er, 
that 
Bh he belt JAE Aaen eel) 
The coefficients of both expansions are equal, but for p >> 5 the 
operators / do not have both times the same order, e.g. for p= 6, 
using both indices: 
A OE EN cpr ENDE TN 
Furtheron we shall no longer use for the / the indices on the 
right of p. 257. 
From (26), (27) and (7) we find the relation: 
m a 
sd A, M; 6: A; M; A dele oe) 
As it is easily proved, that A; M; contains K, just once, the 
coefficients d’;; are idenfical with the coefficient u; in (20) and the 
dij therefore equal to the group characters in the first row of 
Fropenius. We need only know therefore this first row. For the 
case that 4; —=,A; we have 
ai 
dn Re ee Css ZEN 
p 2a 
fon) Wea) 
pms red eS) 
ie 
For more general formulae the cited papers of FRopenius and 
BurnsiDE may be referred to. 
So we have obtained this theorem: 
Principal theorem B. Every affinor of the p'* degree can be expanded 
and for 4;=.2A;: 
