288 UNIVERSITY OF COLORADO STUDIES 



Placing 8=p'' and y = l 



If a? be 80 chosen that 



x+k = (mod p*"-*) 



QP^ will be an operator of order jp^ and may be taken in place of Q. 

 The group G is generated by P and Q, 



PP =1 QP^:::.!. 



As a direct consequence of the foregoing relation the groups in 

 this class correspond to the partition (m — 2, 2) 

 Equation (5) is replaced by 



Q-> PQ=Q^P P l+^p"-* (6) 



as may be seen from congruence (4). 

 From (6) and (3) 



When a and /8 in (6) take all possible values consistent with 

 the defining relations, cases of simple isomorphism between the 

 resulting groups will arise. In order to exclude these all groups are 

 reduced to type forms. 



The groups are now sub-divided under six heads 



1" a and /3 are both prime to p. 



These groups are reduced to a single type in which a and y8 

 are each unity. 



This is accomplished by replacing P^, Q^ by P, Q where x and y 

 are so chosen as to satisfy 



xyS^l (mod p) 



ayH-^fc^^k/Sp-l (mod p'^) 



These congruences follow at once from (7) and admit of solution. 



