C 20=-l)_ c 2(l-a2) 



MILLER— GROUPS GENERATED BY TWO OPERATORS. 239" 



From the last equation it follows that each of the two operators 

 _y 21^2-1)^ _j.2(a2-]) ig invariant under G and therefore it results fronr 

 the first set of equations that 



c 2(^2-1; _ e 2/3(32-1) or <r 2(3-1X3—1) _ T _ f 2(a-lj(a2-l) 



By combining the last equation with 

 it results that 



r 2(a-l);3=-l) T C 2(S-I)(a2-1) 



Jj ^ -^2 



By transforming So~ by s-^''-^'^'^'' and s^- by ^2^^"'~" it is clear that the 

 orders of s^, s^ must divide respectively 



2(/3aa2-i)_ i) and 2(«-'^"') - i). 



The orders of s^, s^ must therefore divide the highest comon fac- 

 tor of 



2{{3-l)(r-—l), 2(a— l)(/3^-l), 2(^^'«=-')-l) 



and 



2(„_i)(a^_l), 2((3-l)(a'-l), 2(a^'^^-^^ — I ) 



respectively. 



The subgroup (H) generated by j^^, s^^ is evidently invariant 

 under G and the corresponding quotient group is dihedral. As the 

 commutator subgroup of H is composed of invariant operators un- 

 der G the fourth derived of G is always the identity. 



§ 2. Groups generated by tzvo operators ivhich satisfy one of the 

 follozving sets of conditions: 



The first set of relations implies that a = /3^ 2. Hence it 

 results from the preceding section that the orders of s^, So must 

 divide 6. If these orders are 6 // is of order 9 and s^s^ transforms 

 the operators of H into their inverses. Hence s-^s^ is of even order. 

 If this order is 2 the group generated by H and s^s^ is of order 18, 

 and it is completely determined by the facts that it contains the 

 non-cyclic group of order 9 and an operator which transforms each 

 operator of this non-cyclic group into its inverse. In this case G- 



