290 REV. T. p. KIRKMAN ON THE THEORY OF 



18. It may be useful to give examples on the theorems 

 D and E. Take N = 5. We should have by theorem D 



— -=6 equivalents groups of 20 substitutions. The sim- 

 4 

 plest of these is made by adding to G = S (i + c) its derivates 



by 2i, 3i and 4^. This gives the group G' 



12345 



24135 



31425 



43215 



23451 



41352 



14253 



32154 



34512 



13524 



42531 



21543 



45123 



35241 



25314 



15432 



51234 



52413 



53142 



54321 



which is found in the Memoir of Cauchy, above quoted, 

 and which has been given by Betti, and first, as I believe, 

 by Galois. I cannot find that any of the equivalent 

 groups has been formed, or that their enumeration has 

 been distinctly affirmed by previous writers. 



We see that the three added groups are derived de- 

 rangements, by writing them thus : 



24135 



31425 43215 





35241 



42531 54321 





41352 



53142 15432 





52413 



14253 21543 





13524 



25314 32154 





The equivalent groups 



of the twentieth order are 





12453 



G'i2534=G'i 





12534 



G'i2453 = G', 





12354 



G'i2354=G'3 





12435 



G'i2435 = G'4 





12543 



G'i2543 = G'5 





all of the form Q G'Qr\ 







In order to determine Q, we compare with G any 



one ff 



of its equivalents, thus : 







5'= 12345 



G= 12345 





24531 



23451 





43152 



34512 





