122 CHAPTER IV. 



Corollary. - - If g = 1 or 2 according as the order of J\I' is 

 equal or is double the order of A(2m, p n ), the number of subgroups 

 of HA(2m, p 2w ) conjugate with A(2m, p 1 *) is 



HA[2m, p 2n ] ~ g A\2m, p n ] = 



where a = 1 ii p = 2, a = 2 if ^> > 2, and q denotes the greatest 

 common divisor of 2m and p n -j- 1. 



134. The conditions that the quaternary substitution in the 



fel === tt \A ~i" ^I 



shall be hyperabelian include the following: 



22 + 81 42 = 



Setting AEEEC^K^ #24^42? we nn d from these conditions that 

 The above substitution then takes the form 



T: 



Inversely, the substitution T is seen to leave absolutely invariant 



if 22; 24; a^ , : 44 belong to the 6r-F[_p 2w ], so that T belongs to 

 H(4, p* ra ). The totality of the substitutions T forms a group G 

 holoedrically isomorphic with the general binary linear group 

 GrLH(2, jp 2w ). Among the substitutions T occur the simple ones of 

 the form 



where ^4 and B are arbitrary marks of the G-F[p* n ] such that 

 We proceed to determine every hyperabelian substitution 



i * i (-!, 



which transforms the subgroup 6r into itself. The product S 



