272 CHAPTER XII. 



enumeration. The largest subgroup of GM(S) transforming G p m d into 

 itself must therefore transform G p m into itself (and thus be a sub- 

 group of H) and transform the groups of the single set of con- 

 jugate 6rrf' ^ amongst themselves. Of substitutions of period p, it 

 must therefore contain only the &. The required group is thus a 

 subgroup of the group H' of order p m K given by the extension of 

 G p m by (r^ 3 ' 0) . Moreover, it is H' itself since any substitution 



of G ( K' Q \ cc such that 2 A = A' is of the [A 1? . . ., A m ], 



0, a 



replaces the elements oo, A by elements oo, A f and consequently trans- 

 forms 6r' into Qr&* . Hence the group ^w d is one of a system of 



conjugate groups. Finally, if the subgroup G' contains no substitu- 

 tion of period p, it is a cyclic subgroup (%*' x) of one of the cyclic 



/nf(, x) 



Or,,! . 

 2;1 



251. Subgroups of G M ( S ) containing operators of period p. - - The 

 substitutions of period p of a subgroup GQ of the G M ( S ] distribute 

 themselves over certain s -f- 1 subgroups Gr^ of the s -f 1 con- 



jugate G^ ( 241). By hypothesis at least one of the orders p ? V 

 is > 1. By suitable transformation within 6rj/( s) , we arrange it so 

 that p m > 1, m = m^ > 0. Under the p m transformers Sp of the G^h , 

 the remaining G^m with m^ > (^ =|= 00), if any, arrange them- 

 selves in sets each consisting of p m conjugate groups. Under the GQ 

 the G ( *m is then one of a set of 1 -f- fp m conjugate groups, f being 



(/*) 



a positive integer or zero. The G$ contains no group G p m^ (m^ > 0) 

 other than the 1 + fp m groups of this set. For, any such group 

 would be one of a set of n^ conjugate groups, where n^ would 

 necessarily have at the same time the forms 1 + f^p m ^ and f^p m . 

 Hence: Every GQ which contains operators of period p contains these 



(M) 

 operators in 1 -f fp m groups G p m conjugate under G$, where for each 



(TQ, f and m are properly determined integers /"> 0, m > 0. 



The groups GQ with f= have been enumerated in 249 250. 

 Consider the group GQ with f^> 1, m > 0. It contains 1 -f fp m groups 



(oo) 



conjugate with a certain G p m formed of the substitutions Si, where 



(oo) 



A ranges over the an additive -group [A 1; . . ., A m ]. The G p m is ( 250) 



(oo) 



self -conjugate within GQ under a certain largest subgroup G p d . Hence 

 250) Q = (1 + fp m }p m d. 



