1898.] MILLER — OX THE QUATERNIOX GROUP. dlo 



If we add ao to the sum of the numbers obtained by means of 

 these formulas we obtain the total number of the substitution 

 groups of degree 2 71 that are simply isomorphic to Q. Among 

 these substitution groups the given regular group is especially con- 

 venient for the study of the properties of Q. 



In what follows we shall, therefore, suppose Q written in this 

 way unless the contrary is explicitly stated. 



It is known that all the substitutions that involve no more than 

 g letters and are commutative to every substitution of a regular group 

 involving the same g letters form a group which is conjugate to the 

 regular group. ^ This conjugate of the given regular group con- 

 tains the following substitutions : 



I ae. bf. eg. dh aceg. bhfd 



agec. bdfh 

 abef, cdgh 

 afeb. chgd 

 ad eh. bcfg 

 ahed. bgfc 



One of the 192 substitutions in these 8 letters that transform 

 one of these two regular groups into the other is the transposition 

 dh. 



The Group of Isomorphisms of Q. 



The largest group in these eight letters that transforms one of 

 the two given regular groups into itself must be transitive, since it 

 includes a regular group. Its subgroup which includes all its 

 substitutions that do not involve a given letter is the group of 

 isomorphisms of Q. We proceed to prove that this is simply iso- 

 morphic to the symmetric group of order 24. To prove this we 

 observe that an operator of order 4 may be made to corre- 

 spond to any other operator of this order in a simple isomorphism 

 of <2 to itself. Hence the first correspondence can be effected in 

 6 ways and the second can evidently be effected in 4 ways, so that 

 the group of isomorphisms must be of order 24. 



This group of isomorphisms may be represented as a transitive 

 substitution group of degree 6, since there are 6 operators of order 



^ Jordan, Traite des Substitutions ^ p. 60. 



