I9I0.] BY TWO OPERATORS. 241 



a given order and this order is an arbitrary even number greater 

 than 2 ; when one of these operators is of order 6 while the other 

 is of order 2 they may generate one and only one group whose 

 order is a given multiple of 12; when both of them are of order 6 

 the order of the group generated by them is a multiple of 36 and 

 they may be so selected as to generate a group whose order is an 

 arbitrary multiple of 36, but each such group is completely deter- 

 mined by its order. 



If we consider all the possible groups which can be generated 

 by two operators satisfying the two conditions under consideration 

 it results from the above that there are exactly three such for every 

 order which is a multiple of 36, one of these is dihedral, another 

 is generated by an operator of order 6 and an operator of order 2, 

 while the third is generated by two operators of order 6. When 

 the order is divisible by 12 but not by 36 there are two and only 

 two such groups, while the dihedral group is the only one that 

 can be generated by two such operators whenever the order is any 

 other even number greater than 2. 



When s^, s^ satisfy the relations s-^~^So"Sj^^=S2'*, sf^s^^So^sf*^ 

 their orders must divide 18 according to the general formulas of 

 the preceding section. The following substitutions prove that these 

 orders may be exactly 18: 



Sj^ = acegihdfh • jk, So = ahidecghf • Im. 



As s^s^ ^^ agd • bell ■ jk • Inv the order of the group generated by 

 these two substitutions is 108. The smallest group that can be 

 generated by two operators of order 18 which satisfy the given 

 conditions is of order 54 and such a group is clearly generated by 

 the two substitutions acegihdfh • jk, abidecghf • jk. 



In general, S{^ is commutative with s.J^ and hence with every 

 operator of H. Similarly, it may be observed that s.,^ is com- 

 mutative with every operator of H. From this it results that every 

 operator of the dihedral group generated by Sj^, s^^ is commutative 

 with every operator of H. It is also clear that these groups can have 

 only the identity in common since the former contains no invariant 

 operator of odd order. It therefore results that G must involve the 



PROC. AMER. PHIL. SOC, XLIX. I95 P, PRINTED JULY 29, I9IO. 



