SUBGROUPS OF THE LINEAE FRACTIONAL GROUP LF(2,pn}. 285 

 For case 1 ) (ii), let E 2 = J, so that EE = J(p n +V/*= - 1. Then 



transforms F 5 into itself and transforms F 2 into 



/ A, SJ e \ 

 \-BJ-\ -A' 



Taking e = 0, 1, . . ., p n , we reach the various p n -f 1 substitutions F 2 . 

 If e be even, the transformer belongs to the hyperorthogonal group 



i 



since J = J . For e odd, it may be given the hyperorthogonal 

 form with determinant a not -square. In fact, there exist in the 



solutions of X*"-^ 1, so that X = -X. Then 



/-E, \ = (Ei 0_\ _ _ (XE, 0_\ = /Xff, _0_ \ 

 \0, JT" 1 / " \0, -E/ " " \ 0, -X^/ ~~\ 0, JO?/ 



of determinant X 2 . Sewce $e groups 6r 60 are all conjugate within 

 G~2M() but form two systems of conjugates within 6rj/( s ). 



260. Summary of the subgroups of GM^), s=p n : 

 s -f- 1 conjugate commutative groups of order s ; 



ys(sl) conjugate cyclic groups of order ^t^-? 2; 1 according 

 as p > 2 ; p = 2 ; 



1 s ~T~ 1 



-r- s (s 1) conjugate cyclic Cr d - for every divisor cZ+ of J|" ; 

 conjugate dihedron 6r2rfx ? for c?q: odd; 



two systems each of M(s)/d+ conjugate dihedron 6^-, for df+ even 



and > 2; 



for p n = Sh 3, one set of M(s)/12 conjugate four -groups; 

 for p n =Sh 1, two sets each of Jf(s)/24 conjugate four -groups 2 ); 



1) *)2 TI _ 1 



4 * each of (2 , 1 f 1)(yt _ 1) conjugate 



commutative groups of order p m , where (2, 1; 1) is read 2, 

 or 1 according as p > 2 with w/& an even integer, p > 2 

 with nfk an odd integer, or p = 2 with n/Jc an integer, and 

 where k is a divisor of m depending on the particular G p m^ 



1) This case may be made to depend on (i) since 5 divides p%n i. Hence 

 each 6? 60 is self -conjugate only under itself within the group GM(S I ) and so 

 within its subgroup 6r2jr(). Hence each 6r 60 is one of a system of 2JT()/60 

 conjugate groups within G%M(S), so that the icosahedral subgroups all form a 

 single system of conjugates within 6r2J/(). They fall into two systems in GTM(*). 



2) For p = 2, the four-groups occur among the groups of order pm= 2* 

 given later. 



