THE HYPERABELIAN GROUP. 125 



linear substitutions in tJie GF[p"* n ] with determinant in the 



The only substitutions of H' which transform G' into itself are the 



substitutions g f of G 1 and the products Vg'. 



136. The substitutions T for which A = 1 form a group G 

 holoedrically isomorphic with the group of binary linear homogeneous 

 substitutions of determinant unity in the GF\p*"\. Since 6r x con- 

 tains T! and T 2 , it follows from 134 that g' and Vg 1 (g f in G 1 ) 

 are the only substitutions of H' which transform G into itself. 

 Hence 6r x is one of a complete set of N conjugate subgroups of H'. 



For p = 2, H' is the simple group HA (4, 2 27Z ) and 6r t is the 

 simple group LF(2, 2 2w ). For p > 2, we pass from H' to the simple 

 quotient - group HA(4, p 2n ~) by making the substitutions T K 106) 

 correspond to the identity. In particular, T_i corresponds to the 

 identity , so that 6r x becomes LF(2,fP*). The only T x belonging 

 to G are T_i and the identity. We have therefore proven the 



Theorem. The simple group HA(4c, p 2t *) contains a complete 

 set of (p* n -r I)p 3n (p n -f l}p n simple conjugate subgroups LF(2, p* n ). 



137. Theorem. The group of hyperabelian substitutions S of 

 determinant unity on 2 indices with coefficients in the GF[p 2n '] is 

 identical with the group of binary linear substitutions of determinant 

 unity with coefficients in the (r-Fjj) 7 *]. 



For m = 1, the conditions 98) and 99) that S shall be hyper- 

 abelian are 



Hence the products <*^a^, a 11 a|" , n ^ belong to the G-F[p*\, 

 being equal to their own (p n ) th powers. Hence if a n =)= 0, the ratios 

 of ff 22 , 21 , cr 12 to n all belong to the GF[p n ~\. Similarly, the 

 products a^ aj", a M f 2 , a 2 2 ft fi a ^ ^ e l n g to tne ^r^|j> w ] and there- 

 fore, if ff 22 =|=0, the ratios of cf ai , 12 , Ojj to 22 all belong to the 

 GF\_p n ]. Finally, if a n = ^ 22 = 0, we have 21 f 2 == 1, so that 

 the ratio of # 21 to 12 belongs to the GF[p*\. In every case, 

 >S has the form 



where the a/y belong to the GF\jP\. Since it is to be hyperabelian 

 and since it is to have determinant unity, we have the respective 

 relations 



Hence, by division, i' w - 1 = l, or a belongs to the GF[p n ]. 



