of order 8p, p being any prime number. 197 



of order 2(p — 1), containing 3 and only 3 substitutions of 

 order 2. These correspond to the substitutions which trans- 

 form those of order Ap in H eye. into their 



2p-l, 2p + l, 4p-l 



powers. We have therefore to examine four types of tails 

 that may be added to H eye, viz. those which are commuta- 

 tive to the substitutions of H eye, and those which transform 

 these substitutions into one of the three given powers. 



Since half of the substitutions of H eye. are the squares of 

 its substitutions there can be only two commutative groups ; 

 viz. the cyclical group (Gi), and the group (G 2 ) obtained by 

 adding to H eye. a substitution (t) which simply interchanges 

 its systems. The squares of the substitutions in the tail of 

 Gr 2 are also the squares of the substitutions of H eye. 



We represent the three substitutions* of the second order 

 which are commutative to t and transform the substitutions 

 of H eye. into the three given powers by s { , s 2 , s 3 . s^, s 2 t, s$t 

 may be used to construct three distinct tails. The first of 

 these contains 2p substitutions of order 2 and 2p of order 4, 

 the second contains 4 of order 2 and 4(p—l) of order 2p, the 

 third contains only substitutions of order 2. We represent 

 the groups containing these tails respectively by Gr 3 , Gr 4 , G 5 . 



Since s x is commutative only with the subgroup of order 4 

 in H eye, and half of the substitutions of this subgroup are 

 squares of its substitutions, there is only one more tail of this 

 type. This contains only substitutions of order 8. Similarly 

 we see that there is only one additional tail of each of the 

 other two types ; and that the former of these contains 4 

 substitutions of order 4 and 4(^ — 1) of order 4p, while the 

 latter contains only substitutions of order 4. We represent 

 the three groups containing these tails respectively by 

 Gr 6 , G 7 , Gr 8 . 



Hence, when p > 2, there are always 8 groups and only 8 

 that contain a cyclical subgroup of order 4p. In what follows 

 we need not consider the groups in which such a subgroup 

 occurs. When p = 2, 2p — 1 and 2p + l are not congruent to 

 1 and 3 respectively with respect to mod 4, as is the case 



* That such substitutions can always be found follows from the fact 

 that we may transform a generating substitution of a transitive cyclical 

 group into any other generating substitution by a substitution whose 

 degree is at least one less than the degree of the group. Since the tirst 

 power of this substitution which is commutative to the group must be 

 contained in the group, its order must be equal to the exponent to which 

 the power into which it transforms the substitutions belongs with respect 

 to mod a,, u being the order of the given eycle. 



