260 CHAPTER XII. 



1680 of type (123) (45) (67), 210 of type (12)(34), 105 of type 

 (12) (34) (56) (7 8), and the identity. The alternating group has sub- 

 stitutions of periods 6 and 15, while G does not. Both groups 

 contain the same number of substitutions of period 7, the same 

 number of period 4, the same number of period 2. But the distribu- 

 tion into sets of conjugates of the substitutions of period 2, or of 

 period 3, or of period 4, differs in the two groups. In particular, 

 G is not isomorphic with the alternating group on 8 letters, each group 

 being simple and of order 20160. 1 ) 



CHAPTER XII, 



SUBGROUPS OF THE LINEAR FRACTIONAL GROUP LI (2, ^). 2 ) 



239. In 108 was defined the group of linear fractional sub- 

 stitutions 



on an arbitrary variable z with coefficients in the G-F[p n ~]. We 

 proceed to represent it as a permutation -group on p n -\-l letters. 

 Suppose e runs through the series of marks of the GrF[p n ~\. For 

 y = 0, z 1 will also run through the series of marks. For y =j= 0, the 



value z = d/y gives 0' = /)> so that z' can not be determined 



as a mark of the field. We may, however, obtain a set of elements 

 which are merely permuted by S by adjoining to the series of marks 



a new element 00=-^? necessarily the same for every mark ji =(= 0, 



since = ^-r = -|p and assumed to combine with the marks A =j= 

 of the field according to the laws 



oo-f Z = A + oo = oo, Aoo = ooA = oo, A/ Qo = 0, oo/A=oo, 



while the indeterminate fraction , ^ is assumed to equal a/y. 



y QO-f d 



Setting henceforth s=jp w , the group LI(2,s) of linear fractional 

 substitutions of determinant unity in the G-F[s] may therefore be 



1) Miss Schottenfels established this theorem by direct calculations, Annals 

 of Mathematics, (2) vol. 1, pp. 147152. 



2) Moore, Mathematical Papers Chicago Congress of 1893, pp. 208 242, 

 Math. Ann., vol. 55 (56?); Wiman, Sweedish Acad., vol.25 (1899), pp. 147; 

 Burnside, Proc. Lond. Math. Soc., vol. 25 (1894), p. 132. The work of Galois, 

 Mathieu and Gierster is cited in the exposition for n = 1 in Klein -Fricke, 

 Modulfunctionen I, p. 411 and pp.419 491. 



