257 
m m 
n= (,4,, — 2,4, + 6 A, + 344, —5 44, + 4inAe 4 Ard 
m m 
\ 
nes as ae 6 
+9 sage “oso ade Ae) u 
6 Es a En = ah = Ee 
o—(*M,, —2°M,, + 6**M, + 34M, —5"*M, + 4 MAM, + 
a a 
(18) 
EE ue OE 
45 M, 49" M, 45° M, HM u | 
All expansions remain valid for special affinors but they are then 
no longer the only possible ones. 
The numbersystem of the class operators. 
Fropenius has shown, that the operators K are commutative both 
mutually and with every permutation /, and that they form a 
numbersystem with & units, which does not contain any nilpotent 
numbers viz. numbers of which some power is equal to zero. As k 
independend units we may choose e.g. K,,..., Kr or A, SA. Aer 
MM, My, which are thus all commutative both mutually and 
with each operator P, A or M. According to the theory of the higher 
complex number systems every system of this kind contains X 
independent numbers /;,¢=1,...,4, the idempotent principal units 
(Haupteinheiten) '), which satisfy the equations 
de foray 
NN . 9 
Fee 0 for iFj 9) 
The sum of the operators J;, which will be called elementary 
operators, is the identical operator 7. When these units are expressed 
in the class operators K: 
tak 
ES S wij K; RRT ee ae AEO 
4) 
and when we write 
K; K; Site IE Di Shy Te” We ce ae 
so we have 
Iek 
Uml = = Umi Umj Lijk « «+ + + + «4 - (22) 
ty) 
from which we see that the coefficients u correspond with the 
group characters x of the symmetrical group as defined by Frosentus’). 
1) See e.g. the author's “Zur Klassifizierung der associativen Zahlensysteme'' Math. 
Ann. 76 (14) 1—66. 
2) Berl. Ber. (96) 985—1021, (98) 505—515, (99) 330—339, (00) 516—534 ; 
see also W. BuRrNsIDE, Lond. M. S. Proc. 33 (01) 146 —162, 34 (02) 41—48, 
1 (03) 117—123), 
