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



by which we see that for every group among those under 

 consideration which is identical with PGP~^, there is one 

 identical with G. 

 Suppose now that 



PGP-i = G, 

 QGQ-^ = G', 

 G' being a different group from G : we have 



QGQ-i = QPGP-^a-\ 



Let us put 



QP = 6'; 

 whence comes 



Q=^P-\ 

 and i = ^P-iQ-i, 



and 6-^=V-^Qr\ 



whence G'=(^GQr^=eGd-\ 



This proves that for every group among those considered, 

 which is identical with G, there is one identical with G', 

 G' being any group equivalent to G. 



We have then this theorem — 



Theorem A. The group G of the order k made with N 



elements and its —r i derived groups are —j— derange- 



ments of G and of its equivalents Gi, G2, G3. • • • • And 

 there are among these derived groups neither fewer nor more 

 derangements of G than of any equivalent G^ of G. 



Cor. If the number of groups equivalent to G is -^— , 



including G, G has no derived derangement, and every 

 derangement of G is a group of permutations different 

 from every derivate of G. 



The same is true of the derived groups and derange- 

 ments of every equivalent to G. 



The number of the equivalent groups and of their de- 



rangemeuts is ( —7— j, which are a system of groups of 



/nN\2 



permutations identical with the system of the \—j—) 



