314 MILLER — ON THE QUATERNION GROUP. [Oct. 7, 



group that contains ^ as a self-conjugate subgroup. Hence we 

 need to consider only one of these three subgroups in connection 

 with tlie study of the intransitive substitution groups that are simply 

 isomorphic to Q. 



We may now state the problem of finding all the substitution 

 groups that are simply isomorphic to Q in the following manner. 

 Such a group contains a transitive constituents of order 8, where a 

 is an integer greater than o. Its other constituents form a group 

 whose order is either 4 or 2. If this order is four these constitu- 

 ents must form the four-group. If it is two these can form only 

 one group for a given set of values off/ and a. Hence we observe 

 that the number of quaternion substitution groups of degree 2 n, 

 « > 3, which contain no constituent group of order 4 is a.^^, where 

 fli is the largest integral value oi x that satisfies the relation: 



n 



X <^ . 



To find the number of these groups that contain a constituent of 

 order 4 we may first find the number of those that contain only 

 one transitive constituent of order 8, then the number of those that 

 contain two such constituents, etc. The sum of these numbers is 

 the number required. Each of these numbers may be directly 

 found by means of the following formula/ in which tV is the num- 

 ber of all the possible substitution groups of order 4 and degree 

 2 «, m is any positive integer, and a^ is the largest value of y that 

 satisfies the relation 



n 



y <^2 



When n = 6 m, N ^= m {^ ni- -^ 6 m -\- 1) -}- a^ 



^ , T,T ^'^ ^^ ^'^' 4- 15 ^^^ + 5) , 



n^=^ d jn -\- 2, N ^^ -i^ 7n {in -{- i) {tti -\- 2) -\- \ -^ a^ 



(2 m -f I) (3 ^fi' ■\- ^^n A^ 4) 

 « = 6 w -f 3, N ^=^ ^ + «! 



;2 =r 6 w + 4, iV^ r=r (w + i) (3 ?;r -f 9 w -f- 4) -[- a^ 



,. 3 {^n + I ) (2 m'' -f 7 m + 4) 

 « = 6 ;;2 -L- 5, N= — — — — -f a^ 



1 Miller, Philosophical Alagazine, 1896, Vol. xli, p. 437, 



