of order 8p, p being any prime number. 199 



We have now found 12 groups of order Sp which exist for 

 every value ofp>2. As the remaining groups cannot con- 

 tain a commutative group of order 4p, they must transform the 

 substitutions of the self-conjugate subgroup of order p accord- 

 ing to a cyclic group of order 4 or 8. Such groups can exist 

 only when p— 1 is a multiple of 4 or 8. We shall examine 

 these two cases separately. 



Groups ivhich exist only ivhen p — 1 is a multiple of 4. 



We shall first consider the case when p — l\s not also a 

 multiple of 8. The substitutions which are commutative to 

 those of the self-conjugate subgroup of order p form a com- 

 mutative group of order 2p. This cyclical group is therefore 

 also a self- conjugate subgroup of the required groups, and its 

 four systems of intransitivity are four systems of nonprimi- 

 tivity of the required groups. Hence we may regard it as 

 the head (HJ of the required groups. 



The tail to these groups contains Ap substitutions which 

 transform the substitutions of the head into a power a which 

 belongs to the exponent 4, mod 2p. We can easily find a 

 substitution s which transforms the substitutions of the head 

 into their a power, and is of order 4 and commutative to t, 

 t representing a substitution of the 4th order which simply 

 interchanges the 4 systems of the head cyclically. H x and 

 st generate a group (Gi 3 ) whose tail contains 2p substitutions 

 of the second order and Ap of the fourth order. 



As s is commutative only with the subgroup of order 2 

 contained in H x there can be only one more group of this 

 type. This (Gr 14 ) may be obtained by using the product of a 

 substitution of the second order in one of the systems of H L 

 into s in place of s. The tail of Gr 14 contains 2p substitutions 

 of order 4, and 4p of order 8. It remains to consider the case 

 when p — 1 is a multiple of 8. 



The preceding 14 groups are all present in this case. If 

 there is an additional group it must transform the substitu- 

 tions of the self-conjugate subgroup of order p according to a 

 cyclical group of order 8. We can easily find a substitution 

 (si) which is symmetrical in the systems of the given self- 

 conjugate subgroup and transforms its substitutions into their 

 /3 powers, /3 belonging to the exponent 8, mod p. 



H 1 and s^ (t being a substitution of order 8 which merely 

 interchanges the given systems cyclically) generate a group 

 (G ]5 ) whose tail contains p substitutions of order 2, *2p of 

 order 4, and Ap of order 8. As the tail of G 15 is not commu- 

 tative to any substitution in the subgroup of order jd, with 



