248 Dr Burnside, On the representation of the simple group of 



In r there are 5 operations A of order two, and characteristic 

 unity, such that ASA = S~^. Moreover F has no subgroup which 

 contains an operation of order 11 and an operation of order 2. 

 Hence P and any one of these 5 operations A will generate V. 



(* = 0, 1, 2, 3, 4), co5=l, 

 the canonical form of S is 



and the most general operation of order two which will transform 

 8 into its inverse, while its characteristic is unity, is 



Now the above invariant, when expressed in terms of the f 's, 

 is a numerical multiple of 



^0^ + 2 (1 + CO + CO') i,i,i, + 2 (1 + co2 + a>3) ^„^,4 



+ (o) + 2a>2) ^1^32 + (^4 + 2a)3) f.fgS + (a>3 + 2a>) ^,^^3 

 + (a;2 + 2co4)f/^2. 



The conditions that the above substitution of order 2 shall 

 leave this unchanged are 



ab^ {w + 2aj2) = aj4 + 2aj3, 

 a26-i (a;3 + 2a;) = a>2 + 2co4, 

 and these equations clearly have 5 solutions. Now a and b must be 

 rational functions of a and co, for otherwise the group could not be iij 

 one of finite order. They may be expressed in that form as follows ''i 

 Put 



a + a-^ = XQ, a2 + a-2 = Ai, a^ i- a-' = X^, 

 a3 + a-3 = A3, a^-\-a-^ = X^, 



{i = 1, 2, 3, 4). 

 Direct multiplication then gives 

 ^1^4 = /^2/^3 = (^ + 2co2) (w2 + 2a>4) (a)3 + 2a;) (w* + 2aj3) = 11, 

 /zi2 = (a, + 2a>2)(a;2 + 2a>*)^2, 

 ^2^ = (^2 + 2a>4) (a;4 + 2a)3) ;,^, 



/X42 = (a>4+2a>3)(a>3 + 2w)/X3, 

 ^32 = (a;3 + 2a>)(a) + 2a;2)^j. 

 From these results it follows at once that 



a = ^^3 ._ /xi 



(o) + 2a;2) (co3 + 2a;)' (a; + 2a;2) (a;2 + 2a;^)' 



involving 



a-i =. — 1^ 7,-1 fH 



(a;2 + 2a;*) (a;* + 2a;3)' " (oj^ + 2a;) (a;* + 2a;3) 



I 



