240 MILLER— GROUPS GENERATED [April 23, 



is of order 36 and such a G can clearly be generated by the two 

 substitutions 



s^ = abc • de, So^=ab ■ def. 



As we may annex to these substitutions any constituents of order 

 2 in new letters it results that the order of s^s^ is an arbitrary even 

 number and hence the order of G is an arbitrary multiple of 36. 

 To prove that there is only one such group of a given order the 

 following considerations are helpful. The cyclic group generated 

 by (^1^0)- has only the identity in comomn with H since i-^^g trans- 

 forms each operator of H into its inverse. This cyclic group is 

 invariant under G. In fact each of its generators is transformed 

 into itself by half the operators of G and into its inverse by the rest 

 of these operators since (SoS^)~ =^ (.y^^yo)"- = V^V^V^V^ = -^2%"^ • 

 s-f'^Sj-'^s.^-'^s-^'^ = -^2-^"i~^'^2-^i"^- It is now clear from the general theory- 

 of simply isomorphic groups that if s^, s^ ; ^Z, s.,^ are two pairs of 

 operators satisfying the given conditions and if they generate groups 

 of the same order these groups are necessarily simply isomorphic. 

 Hence the theorem : // each of tzvo operators of order 6 transforms 

 the square of the other into its fourth pozvcr these operators may 

 be so selected tliat they generate a group zvhose order is an arbi- 

 trary multiple of 56 and there is only one group of each such order 

 zvhich can be generated by tzvo of its operators of order 6 satisfying 

 the given conditions. The second derived of each of these groups 

 is unity. 



When the order of only one of the two operators s-^, s^ is 6 that 

 of the other must be 2, since it transforms the square of the former 

 into its inverse. In this case H is the cyclic group of order 3 

 and the order of G is an arbitrary multiple of 12. The second 

 derived of all of these groups is the identity since (^1^0)^ ^'^^ ^1^ 

 generate an invariant abelian group and the corresponding quotient 

 group is the four-group. When the order of s-^ is 3 that of .Jo mvist 

 be 2 and G is the symmetric group of order 6. If s^, So are both 

 of order 2 they may be so selected as to generate an arbitrary 

 dihedral group. Combining these results we have that when s^, Sr, 

 are both of order 2 they may generate one and only one group of 



'Bulletin of the American Mathematical Society, Vol. 3 (1897), p. 218. 



