198 Dr. G. A. Miller on the Operation Groups 



for the other values of />. Hence some of our remarks do not 

 apply to this case. In fact there are only 6 groups of order 

 16 which contain a cyclical subgroup of order 8. 



Groups containing H. 



By adding t to this head we obtain the third and last 

 commutative group (G 9 ) of order Sp. There are, therefore, 

 three and only three commutative groups of this order for 

 every value of p> 2. When p=2 there are five such groups. 

 The tail of G 9 contains 4 substitutions of order 2 and 4(j9 — 1) 

 of order 2p. The squares of these substitutions are clearly 

 the same as the squares of those of H. It remains to find 

 the non-commutative groups that contain H. 



The isomorphic group of order 8 contains at least three 

 substitutions of the second order. If this group is commuta- 

 tive the corresponding tail must transform the substitutions of 

 H into their — 1 powers, as 2p has primitive roots*. We can 

 easily find a substitution (/) of the second order which trans- 

 forms one of the 4 cycles of a substitution of order 2p in H 

 into its —1 power. By making s f symmetrical in the ele- 

 ments of the other cycles of the same substitution we obtain 

 a substitution (s) which evidently transforms H into itself. 



The tail of the group (G 10 ) generated by st and H contains 

 only substitutions of the second order. Since s is commuta- 

 tive to the substitutions of the second order in H, we may 

 construct a group (G n ) whose tail contains only substitutions 

 of the fourth order by using, in place of s, the product of a 

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

 into s. The other two groups which may be constructed in 

 the same way as Gr n are conjugate to it with respect to (s 2 ), 

 the pth. power of two cycles commutative to t, these cycles 

 being contained in some substitution of H whose order is 2p. 

 It remains to examine the case when the isomorphic group of 

 order 8 is not commutative. 



Since this group of order 8 contains at least 3 substitutions 

 of the second order and is non-commutative, it is fully deter- 

 mined. The corresponding tail must interchange two of the 

 divisions of the head and transform the substitutions of the 

 other two divisions into their —1 powers. The group (G 12 ) 

 generated by H and ss 2 t clearly satisfies these conditions. Its 

 tail contains 2p substitutions of order 2 and 2p of order 4. 

 The other possible group is conjugate to G 12 with respect 

 to a substitution of the second order in one of the systems 

 of H. 



* Cf. Serret's Cours d'Algebre Superieure, vol. ii. p. 82. 



