242 CHAPTER XI. 



OHAPTEE XL 



OPERATORS AND CYCLIC SUBGROUPS OF THE SIMPLE 

 GROUP LF(3, p n ^) 



224. By 108 the group G = LF\Z,p n ) of all substitutions of 

 determinant 1, 



8: x'=^- 



in which the coefficients # belong to the 6rJP[^) n ], is a simple group 

 of order 



where <Z is the greatest common divisor of 3 and p n 1, so that 



d = 1, if p n => 3 or 3? - 1; d = 3, if # 3Z + 1. 

 The equation T 3 = 1 has in the G-F[p n ] a single root = 1, if d = 1; 

 but has three roots 0, 6 2 , 3 ^ 1, if c? = 3. Hence, if e2 = 1, there is 

 a single homogeneous substitution of determinant unity 



Z: |5 = a |i + / 2 | 2 + ,-3i 3 (* = 1, 2, 3) 



which, when taken fractionally, leads to the non- homogeneous sub- 

 stitution 8. If d = 3, let denote the homogeneous substitution of 

 determinant unity which multiplies each index by 6. Then there are 

 exactly the three homogeneous substitutions of determinant unity, 

 Z, 0Z = I0, 2 ZEEZ0 2 : 



0T: gj - 6 r (ali + o^fei + ,- 8 |) (* = 1, 2, 3), 



which, when taken fractionally, lead to the non -homogeneous sub- 

 stitution 8. Combining the two cases, we may employ the group 

 of ternary linear homogeneous substitutions of determinant unity in 

 place of the group 6r provided we consider to be identical the d sub- 

 stitutions Z, 0Z and 2 Z. Under this convention concerning the 

 homogeneous substitutions, we employ henceforth the homogeneous 

 notation for the substitutions of the group 6r. 



225. Any substitution of 6r can be reduced by a linear ternary 

 transformation of indices (not necessarily in the GF[p n ] and not 

 necessarily of determinant unity) to one of the canonical forms A y 

 jB, (7, D, E of 222. In the present case, the determinants of 

 A, . . ., E must be unity. 



1) For n = 1, Burnside, Proceed. Lond. Math. Soc., vol. 26, pp. 58106; 

 for general n, Dickson, Amer. Journ., vol. 22, pp. 231252, where certain errors 

 in Burnside's paper are pointed out. 



