302 CHAPTER XIII. AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 



Corollary. The quotient- group LF(2,p n ) is Jioloedrically iso- 

 morphic with the abstract group F generated ~by the operators T and Si 

 subject to the relations T 2 = I togetlier with a) and c). 



279. For A = or 1 or for ft = or 1 , relations c) always 

 reduce to d) upon applying a) and b). For the group LF(2,p n \ 

 d) becomes 

 D) ft r) = I. 



If neither A nor p is or 1, the product of any two consecutive 

 subscripts in c) is not unity, the first subscript A being regarded as 

 consecutive with the last subscript ({* l)/(Aft 1). Using any two 

 consecutive subscripts as the initial A, ft, the resulting identity c) is 

 seen to be an immediate consequence of the given identity c). Taking 

 for I any one of the p n 2 marks =(= 0, 1 and for ^ any of the p n 3 

 marks =[= 0, 1, A" 1 , the remaining subscripts in c) are different from 

 and 1. Hence those identities c) which do not reduce to D) are 

 equivalent in sets of five, an exception being those with all subscripts 

 equal to A, where A 2 -j-A=l. If the latter has 6 solutions in the 

 GrF[p n '\, it follows that there are exactly 



distinct identities c) not immediately reducible to D) . For p = 2, 

 6 = or 2 according as n is odd or even; for p = 5, tf=l; for 

 p =)= 2, 4= 5, = or 2 according as p n = 5k 2 or p n = 5k 1. 



280. For the group LF(2, 5) of order 60, the N= 2 relations c) are 

 (S 2 T) 5 = J, S 2 IS* TS B TS 3 TS T = I. 



These may both be derived from a), D) and T 2 = I, so that LF(2, 5) 

 is generated by A^S-L, ~B = T subject to the relations 



282) J. 5 = 7, 2 = I, (AB)* = I. 



In proof, we apply D) repeatedly and find that 



Hence also (T$ 3 ) 5 = J, so that the second relation becomes 



TS, T = S 1 (S, T^S, (S L T) 2 = 



281. The group LF(2, 2 2 ) of order 60 may be generated by 

 A = TS { and B = S^ subject to the relations 28*2), where i and i* 

 are the roots of x 2 -f- # = 1 (mod 2). Indeed, the N=G = 2 

 relations c) to be considered in addition to D) are 



