79 



Def. If PG and GP are the same group of permutations, 

 PG is a derived derangement of G. 



(8) Theo. A. The ^ derived groups of G (including G), 



Hi 



are derangements of the groups G G^Ga * * G^ equivalent to 

 G ; and there are among these derived groups neither more 

 nor fewer derangements of G than of any group equivalent 

 to G. 



(9) Cor. If the number of groups equivalent to G is 



^-—^ G has has no derived derangement, except itself; and 

 n 



every derangement of G is a group of permutations different 

 from every derived of G. The same thing is true of the 

 derangements and derived groups of every equivalent of G. 



If the number of groups equivalent to G be fewer than that 

 of its derived groups, we know that G has derived derange- 

 ments. 



If M be the number of groups equivalent to G, -^^^ is the 



number of derived derangements of each (G^) of these equiva- 

 lents (including (G^) in this number), which are found among 

 G and its derived groups. 



Each of the equivalent groups G Gi . . . G^ forms with its 

 derived derangements a different group (containing (1)) of 



-TTT- substitutions. 

 M 



I believe that this theorem A (8) and the corollary (9) that 



follow it, are among the faits noiiveaux of my Memoir. 



They lead directly to all the enumerations and extensions 



following. 



(10) Theorem B. The number of different groups of the 

 order N (1, A, A^ . . Af^-^) made with N elements, which 



are N powers of one substitution, is ^ ^^~ ^ , R^ being the 



number of integers less than N and prime to it, unity 

 included. 



