200 Operation Groups of order 8p, p being any prime number, 



the exception of identity, there can be only one group of this 

 type. Hence there are always 12 groups of order 8p (p>2) 

 which contain a self-conjugate subgroup of order p ; when p — 1 

 is a multiple of 4 or 8 there are respectively 14 or 15 such 

 groups. It remains to consider the 



Groups of order 8p which do not contain a self-conjugate 

 subgroup of order p. 



When p=7, the equation 



kp + 1 r 



is satisfied by k=-b = l as well as by k = 0, b = S. If the sub- 

 group of order p is not self-conjugate there must be 8 such 

 subgroups. These contain 8 x 6 = 48 substitutions besides 

 identity. The subgroup of order 8 must therefore be self- 

 conjugate, and its 7 systems of intransitivity must be systems 

 of nonprimitivity of the required groups. 



Since the substitutions of the group (H 2 ) of order 8 cannot 

 be commutative to the entire group, they must be trans- 

 formed according to a group of order 7. Hence all these 

 substitutions are of the second order, and H 2 is fully deter- 

 mined. If we add to H 2 a substitution (r) of order 7 which 

 simply permutes its 7 systems cyclically, we obtain a group 

 (Gr 16 ) whose tail contains only substitutions of order 7. As 

 no substitution of H 2 , besides identity, is commutative to t, 

 there can be no other group of this type. 



Hence there are 13 groups of order 56 ; 12 of these con- 

 tain a self-conjugate subgroup- of order 7. The remaining 

 one contains 8 conjugate subgroups of order 7 and a self- 

 conjugate subgroup of order 8. The last group occurs for 

 the first time as a group of degree 8*. 



The only other value of p > 2 for which there can be groups 

 which do not contain a self-conjugate subgroup of order p 

 is 3. In this case it is known that there are three such 

 groups t- Hence all the groups of order 8/? are completely 

 determined. 



Paris, June 1896. 



* Cf. Cole, ' Bulletin of the New York Mathematical Society,' vol. ii. 

 p. 189. 



t Cf. Levavasseur, Comptes Rendus, March 2, 1896. 



