124 CHAPTER IV. 



If every a^ = 0, when i and j are both even or both odd, for 

 p n > 2, $ reduces at once to the form 



This is of the form Vg, where g is of the form 109) and V denotes 

 the hyperabelian substitution not in 6r, 



' ri^ 1 ~ ?2? ^2 == *1? 3 == ~ *4? fe4 == b3' 



The theorem stated below has thus been proven for p n > 2. 



For j} w > 2, we consider the reciprocal of S and find the 

 conditions corresponding to 111) and 112) that S~ 1 shall transform 

 Tj_ and T 2 into substitutions belonging to G, viz., 



114) tfijtf4ifc=0, ff 2 iff3jfc=0 (, & both even or both odd). 



By 111), 112), 114), S must be of the form g or Vg, g being of 

 the type 109). To illustrate the method of proof, let a ls =j= 0. Then 

 41 = a 43 =0 by 114). Since oc^ and 44 can therefore not both 

 vanish, a 12 = 14 =0 by 114). Likewise from 111) c^ 12 = 32 =0, 

 23 = cc 43 = 0. The hyperabelian condition involving the coefficients 

 of the first and third rows then gives 13 J 4 i== 0, whence a M = 0. 

 Then 31 and 33 can not both vanish, so that tt 21 =0 by 114). 

 Hence S has the form 109). 



The order of G is (> 4 n - 1) (p* n p 2n ) by 99. The order 

 of JT(4,_p a ) is (p* n I}p 5n (p 3n + I)p 2n (p 2n - l)p n (p n +l) by 132. 



Theorem. Tlie quaternary hyperabelian substitutions T with 

 coefficients in the CrF[p 2n ] form a group G holoedricalty isomorphic 

 with GLH(2, p 2n ~). The only substitutions of H(4, p 2n ) which trans- 

 form the subgroup G into itself are of the form T or VT. jff(4, p n ~) 



contains exactly JVn: - (> ~(p 5n -j- I)p 3n (p n + ^)p n subgroups conjugate 

 with G. 



135. Consider the subgroup H' formed of the substitutions of 

 H(4,p 2n } of determinant unity. By 132, its index is p n +l. The 



determinant of the substitution T is seen to equal A~^ . Those 

 substitutions T in the GF[p 2ri ] whose determinant is unity form a 

 group G' of order (p** T)p 2n (p n 1). Since T and T 2 are of 

 determinant unity, the proof in 134 leads to the following theorem: 



Within the group H' of quaternary hyperabelian substitutions in 

 the GF[p 2n ] of determinant unity, the subgroup G' of the substitutions T 

 of determinant unity forms one of a complete set of N conjugate sub- 

 groups, each being holoedrically isomorphic with the group of binary 



