318 MILLER — 0>r THE QUATERNION GROUP. [Oct. 7, 



substitutions these substitutions must form a subgroup of this group 

 of isomorphism and the given equation must assume 24 -^ a dif- 

 ferent forms which are equally true in case it is transformed by all 

 these substitutions, e. g., each of the three equations in the last set 

 given above admits a cyclical subgroup of order 4. Hence each 

 of these equations gives rise to 24 -f- 4 = 6 true equations. In 

 addition to the three that have been given we have ( — if= {—Jf^= 



We have already noticed that the group of cogredient isomor- 

 phisms of ^ is the four-group. Hence Q has only two operators 

 that are commutative to each one of its operators. ' These are evi- 

 dently the operators which correspond to i and — i in the quater- 

 nion unities. These two unities are therefore the only ones in the 

 quarternion number system that are commutative to all the numbers 

 of the system. It need scarcely be remarked that any one of the 

 three cyclical subgroups of order 4 contained in Q may correspond 

 to the unities of the ordinary complex number system. 



Relation Between the Quaternion Group and the Hamil- 



TONiAN Groups. 



One of the most remarkable properties of the quaternion group 

 is that each of its subgroups is self-conjugate. Dedekind has 

 called all the groups which have this property Hamiltoniaji groups 

 and he has pointed out that the quaternion group is of fundamen- 

 tal importance in the study of the Hamiltonian groups.^ It has 

 recently been proved that every Hamiltonian group is the direct 

 product of an Abelian group of an odd order and a Hamiltonian 

 group of order 2", and that there is one and only one Hamiltonian 

 group of order 2" for every integer value of a greater than 2? 



It is easy to see that the direct product of the quaternion group 

 and the Abelian group of order 2""^ which contains 2""' — i 

 operators of order 2 is Hamiltonian. Since there is only one 

 Hamiltonian group of this order it follows that every such Hamil- 

 tonian group may be constructed in this manner. Hence we have 

 that every Hamilto7iian group whose order is divisible by 2", but 

 not 2""^^ t7iust be the direct product of some Abelian group of an odd 

 order, the Abelian group of order 2""^ which contains 2*^"^ — i 

 operators of order 2, and the quaternion group. 



1 Dedekind, loc. cit. 



2 Miller, Comptes Rendus, 1898, Vol. cxxvi, p. 1406. 



