SUBGROUPS OF THE LINEAR FRACTIONAL GROUP JLF(2,pn). 265 



If A= 0, then JB = 1, so that F= - Th e latter trans- 



forms Q 9 into ( ! ) = Q~ g , which is distinct from Q y unless the 



latter be of period two. 



The largest subgroup of H within which the cyclic group G 9 +\ 



2;1 



is self - conjugate is therefore a dihedron- group of order 2 -57-7- 

 Hence JET, and consequently also 6rjf(,), contains 



s (s 2 - l) . 9 s -fi ___!_/ -, \ 

 _______ j __ == s(s- 



cyclic groups conjugate with 6r,+i. Each of these is defined by any 



2;1 



substitution lying in it (the identity excepted) as the largest cyclic 

 group containing that substitution. The s (s 1) groups have there- 

 fore only the identity in common and contain in all s (s I) 2 or 

 -y s 2 (s 1) further substitutions according as p > 2 or p = 2. 



244. To verify that we have now enumerated all the individual 

 operators of GM(S) and consequently all the largest cyclic subgroups, 

 we note that 



It was shown that if any substitution S of a cyclic Gr s +i be of 



2;1 



period > 2 (viz., S 4= $"*), then $ is transformed into itself by no 

 substitutions of G M () other than those of the cyclic G t +1. Hence 



2;1 



the latter is the largest commutative subgroup of GM(S) which con- 

 tains the substitution S. A commutative subgroup containing an 

 operator of period > 2 and different from p is therefore a 

 cyclic group. A commutative group containing an operator of 

 period p contains only operators of period p ( 241). Hence if a 

 commutative subgroup of GM()> P> < %, contains an operator of period 

 > 2, it contains at most one operator of period 2. 



245. Cyclic and dihedron groups and their subgroups. The abstract 

 dihedron -group 6r 2ifc may be generated by operators A, ~B subject 

 only to the generational relations 



