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MILLER — OX THE QUATERNION GROUP. 



[Oct. 7, 



4 that can be made to correspond and these generate Q. As this 

 substitution group cannot contain a substitution whose degree is 

 less than 4 and the transitive groups of degree 6 and order 24 that 

 have this property are simply isomorphic tg the symmetric group of 

 this order it follows directly that the group of isomorphis?7i5 of Q is 

 the symmetric group of order 24 and that the group of cogredient isc- 

 vi07phisms is its s e If -co7tj agate subgroup of order 4. 



There are two transitive groups of degree 6 that are simply iso- 

 morphic to the symmetric group of order 24. In one of these the 

 subgroup which contains all the substitutions that do not include a 

 given element is the cyclical group of order 4 while in the other 

 it is the four-group. It remains to determine which of these two 

 groups is the substitution group of isomorphisms of Q. This may 

 be easily done by making Q simply isomorphic to itself in the fol- 

 lowing manner : 



The substitution which corresponds to this isomorphism is given 

 by the second columns of letters ; hence it is bdfh and the substi- 

 tution group of isomorphisms of Q is the one which Prof. Cayley 

 represents by (=t abcdefja^-^ 



It is known that Q is simply isomorphic to the eight unities 

 (i, — I, i, — i, j, — -j, k, — k) of the quaternioji mwiber system. As 

 Q can be made simply isomorphic to itself in 24 different ways the 

 simple isomorphism of Q to these unities or of these unities to 

 themselves may also be written in 24 ways. The following is one 

 of these ways : 



ae. bf. eg. dh 

 aceg. bdfh 

 agec. bhfd 



It may be very easily verified that the following relations are 

 1 Quarterly foinnal of Mathematics, 1891, Vol. xxv, p. 80. 



