Functions given by Groups. 329 



a subgroup of G^^, we can write 



G-d = J«H-1 + PJm+l + P2jm+1 + . • . + Pi-lJm+1 J {b) 



where l.{in-\- 1) = L is the order of G^^, which under 2 is 



here by PeJm+i is meant what PeJ^+i becomes under S. 



It is plain that, in I,+i under its S, all the t-\- 1 products 

 are algebraically identical. Wherefore 



5m+i = (^«+i)Po {a) 



is the first term on the right of {c\ Ipo being unity under2. 

 To find by an example the other terms of (^), let 



;2 = 4, S = aa/3/3, L = /. (w + i ) = 2 '4, 

 and let 



Gd=i234 34i2 = Jm+i + PJm+i; (Pi = 34i2). 

 2143 4321 

 1243 3421 

 2134 4312 



Then we have 



aa/5/3 aa(3l3 



6d=I234 + 34I2=3'm+l + Pl5m+l, W 



+ 2143 + 4321 

 + 1243 + 3421 

 + 2I34 + 43I2 



le. G.- 4-(i"2«3 V + 3 Vi^2^) = (m+ i)(lp, + pj, (/3>o). 



or every term in Gd is (m+i) times repeated ; so that, if 

 Gd has L terms, the number of different products in the 

 function (3d is always L : (m+i), (in'^o) ; and every value 

 of (Bd has the same number of products. 



It is not to be supposed that, in obtaining the equivalents 

 in (A„) that we compare, we have to operate on entire 

 columns of substitutions. In all cases the groups of our 

 table (A„) can be labelled, in many cases by three substitu- 

 tions at most, and frequently by one (/) only, so that, instead 

 of the column dGQ~\ dld~^ in one line often suffices for 



