1898.] MILLER — OX THE QUATERNION GROUP. 313 



notation we represent Q as a. regular substitution group in the fol- 

 lowing manner : ^ 



I ae. bf. eg. dh aceg. hdfh 



agec. hhfd 

 abef. chgd 

 afeb. cdgh 

 adeh. bgfc 

 ahed. befg 



The Different Ways in Which Q May be Represented 

 AS A Substitution Group. 



We observe, in the first place, that Q cannot be represented as a 

 non-regular transitive substitution group. If such a representation 

 were possible Q would have to contain some subgroup of a prime 

 order that is not self-conjugate." As it contains only one subgroup 

 of order 2 this must clearly be self-conjugate. Hence we observe 

 that there is only one transitive substitution group that is simply iso- 

 morphic to Q. 



It is known that the number of the intransitive substitution 

 groups that are simply isomorphic to a given group is an increasing 

 function of the degree, which becomes infinite when the degree 

 becomes infinite. We proceed to determine the nature of this 

 function in the present case. Since every group whose order is the 

 square of a prime number is Abelian, a substitution group which is 

 simply isomorphic to Q must contain at least one transitive con- 

 stituent of order 8 and its degree must be 2 71, n being a positive 

 integer greater than 3. 



We have seen that Q contains only one subgroup of order 2. 

 With respect to this it is isomorphic to the four-group, since this 

 subgroup contains the square of each one of its operators. As a 

 subgroup whose order is one-half of the order of the entire group 

 must always be self-conjugate, Q contains three self-conjugate sub- 

 groups of order 4. Since none of these three subgroups is charac- 

 teristic^ they must be transformed into each other by the largest 



1 Cayley, Quarterly Journal of Mathematics, 1891, Vol. xxv, p, 144. 



- Cf. Dyck, Alathematische Annalen, 1883, Vol. xxii, p, 90, It may be 

 remarked that the statement on p. 10 1 of this article that a group which can be 

 represented only in the regular form contains only self-conjugate subgroups is 

 not quite correct, as may also be inferred from other parts of the same article. 



3 Frobenius, Berliner Sitziingsberichte, 1895, p. 183. 



