Functions formed on Groups. 37 



being unity ; and the number of different equivalents of 

 G rf is /. We proceed with S = afiyyll. 



7. In the case before us we take the above G d = (356142), 

 which has in common with 1, +1 the sub-group (art. 6) 



J2 = 1 + O, and we have 

 H 2 - 1 + A. 

 Wherefore 



AGrfX"^ 1 24356 -356142 -124356 = (45 1632) = G e 



and this G, is to be marked out of our table of 60 equiva- 

 lents, because 



We have here no use for 6 in I (+1 , for it gives 



0G.6- 1 = XeGrfO^A- 1 = \Ga\~ 1 . = G e 

 just found. 



Thus it suffices, as was promised in (F.G. 13), to use 

 only the factor-group H 2 , and there is no inelegance of 

 repeated outmarking. 



Taking next G 2d = (542631), we obtain, by the same Ho, 



G 2<! = \G2 (J \~ 1 = (635i24) or its reciprocal (452631), 

 which is also to be marked out. 



We retain G d and G id , which have, under 2, 60 values each 

 of 6 terms (F.G. 13) ; unless the reader can find a C = K in 

 I, +1 , which makes (a) art. 3, true of one or of both. 



There remain 60 — 4=56 of our 60 groups which have 

 no 9 in common with I, +1 , which now is (m = 6) 



I«+i =I s +i= I2345 6 = H 3+1 

 124365 = 6! 

 123465 = 63 

 124356 = 63 



Taking as G D 234561, we form 



VJaflT 1 - 246315, 6,G D 6^ = 234615, 6 3 G D 6 3 - 1 = 245361, 

 which are G El , G E . 2 , G e3 , and are to be marked out, as all 

 giving the function and value 6+' ■ under a(lyy$& 



