AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 293 



The abstract group 6r 60 generated by E ,E 2 , E 3 subject to the relations 

 271) E* = E; = ES = I, (E 1 E 2 ^ = (E 2 E^ = (E 1 E 3 y = I 

 is put into holoedric isomorphism with Z/ 60 by the correspondences 



272) jsi~(5j,y, # 2 ~(i 3 i; 4 )(u a ), # 3 ~(i 4 i 5 )(y 2 ). 



The following theorem is quite evident: 



The abstract group 6r 16 generated by B lf B 2 , B 3 , _Z? 4 subject to 

 the relations 



273) Bf = I, BtBj^BjBi (i, j = 1, 2, 3, 4) 



is put into holoedric isomorphism with L i& by the correspondences 



274) B^C^C,, B 2 ~C 2 C B , J0 8 ~<7 8 C 4 , # 4 ~<7 4 C 5 . 



If we impose the relations 275) below, the two groups 6r 60 and 

 6r 16 will be permutable. Writing the analogous relations between 

 the corresponding orthogonal substitutions 272), 274), we readily see 

 that they are satisfied. We have therefore the theorem: 



The abstract group generated by E , E 2 , E s , B lf B%, B 3 , B subject 

 to the generational relations 271), 273), and 



E i 



275) < 



is of order 960 and is holoedrically isomorphic with the linear group A$ 6Q . 



271. Theorem. - The abstract group 6r 960 of 270 may be 

 generated by the operators E ly E 2 , _E 3 , S 1 subject to the generational 

 relations 



These relations foUow immediately from 271), 273), 275), with 

 the exception of (B^E^) 3 = I, which is derived from the first two of 275): 



E ~BE 1 = B, 



E~ B 1 E 3 = B 1 , E 5 B 2 E 3 = B 1 B Z , E 3 



B 1 E 1 B 1 = B 2 = E^B^E^ , 



together with Ef = B* = I. Furthermore, we have by 275), 

 277) Bi = E 



