180 



CHAPTER VII. 



Since Q is holoedrically isomorphic with J.(4, p n ), its order is, by 

 115, half of that of 0^(5, p). To complete the proof of the 

 theorem, we then show that Q is holoedrically isomorphic with a 

 subgroup of $i(5,p w ) containing all the generators of the group 

 OJ(5,jp") defined in 181. 



Let first 1 be the square of a mark i of the GI^p**]. Set 



162) 

 whence 



Hence G is holoedrically isomorphic with 1 (5,jp n ). Proceeding 1 ) as 

 in 187, we find that Q 1 (Q expressed in the indices | t -) contains 

 the substitution Of' 4 if and only if it be a 3,4, also that Q' contains 

 ( 3 5 6 ). The latter with (i 2 i 3 5 4 ) will generate all even substitutions 

 on | 2 , . . ., le hy the preceding section. But (S 2 S 8 ? 4 ) expressed in the 

 indices Y;J is 



12 



13 



-14 



23 



-24 



34 



This is seen to be the second compound of the following special 

 Abelian substitution: 













 



n i 







-0 











It follows that $' contains every ($'/ (', j = 2, . . . 6; i 4= j)- Also 

 $' contains OJjOgS and hence every OfifOfcf. For _p n =5, we 

 take i = 2. Expressing the additional generator R 345 in the indices 

 Yijy we reach the substitution (mod 5) 



1) Comparing the transformations of indices 154) and 162), we note that 

 they are identical as far as | 8 , fj 4 , 5 and | 6 are concerned. 



