274 CHAPTER XH. 



Comparing this maximum with the actual number fp m (p m 1) ; we 

 nave p m 1 < (2; 1) d. Since each of the corresponding (2; l)d 

 marks q must be one of the e marks a t - of 249 , then (2; l)d^e, 

 where e = (2, 1; 1) (p k - 1) Finally (70), k is a divisor of m. 

 Hence 



252) ^-1^(2; 1)^(2, l 



Since the third number is always <; 2 (p k 1) < 2jp* 1, we have 

 p m <2p k , so that m = It, m being a divisor of k. The additive- 

 group [/L 1? . . ., AJ is therefore its own multiplier CrF[p k ] and every 

 A is zero or a multiplier %. 



There are in all two cases: 



[A] ro-fc, !*-l=(2;l)d, Q = (1 



[B] m = ~k, p>2, n/Jc even, p* - 1 = d, Q = (1 



where for (2; 1) we read 2 or 1 according as p > 2 with nf~k odd 



or p = 2. 



The following lemma finds repeated application below: 



If Vj (y$ =(= 0) be of period 2, the ratio &jlyj differs from the 



of every other V f and so is a characteristic invariant of the V^ 2. 

 For i =%=j, ViVj is not of the form V, h i, since otherwise 



contrary to the choice of the extenders Ff. Hence in V,-Yj the term 

 corresponding to y is =j= 0, viz., a^ -f- y* ^y H= '0 . Dividing by 

 and applying 8j = a, (Vf being of period 2) we find that 



*/y< - Vyy 4= - 



252. For case [A] w^ j5 fc > 2, ^e ^rowp G^Q is the group G M ( P ^ 

 of all linear fractional substitutions of determinant unity in the CrF[p k ]. 

 For p k = 2, G-Q is a dihedron 6r 2 (i+2/), which for /*=! is the G-M&)- 



For p k > 2, it is shown that every Vj may be chosen so that 

 a j> $h Yi> dj all belong to the GrF[p*]. Hence G-Q is a subgroup 

 of G-M(ifi). But, if f> 1, Q > M O*). Hence must /"= 1, Q Jf (_p*), 



SO that GTQ = G'M(p k )' 



For case [A], relations 252) become equalities, so that the earlier 

 argument shows that, for Vj and rj given (3^ =j= 0, 97 =|= 0), there 

 exist exactly (2; 1) marks A of the [^, .-, ., AJ which satisfy 251). 

 The given 77 may be any one of the multipliers x, since the number 

 (2; \)d of i?'s equals the number p k 1 of >c's. 



The extender Vj may be replaced by any one of the products 

 VfyiVj and in particular by one of period p, having therefore 

 K j + $s = 2. Changing if necessary the signs of all four coefficients 



