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



demonstration may be seen in his "Memoire sur les 

 arrangements que Ton pent former avec des lettres don- 

 nees," Exercises d' Analyse et de Physique Mathematique, 

 tome troisieme. And its truth will be more simply evident 

 by what follows on the construction of groups. 



§2. 



Deranged groups : Derived groups. 



6. Let G be a model group of k substitutions 

 lAjAg- •A}c_i, 

 and let P be any substitution not found in G. 

 We can write the product 

 GP = P 

 AiP 

 A,P 



AP 



in a vertical column. None of the k arrangements thus 

 made is in G ; for if 



we have 



A,™ r — A»i 



P = A-^A„=A^, by (2); 

 that is, P is one of the substitutions of G, contrary to 

 hypothesis. The only effect of P on G is to change the 

 arrangement of entire vertical columns of G, as is evident 

 if we compare together the products 



(i)P, AiP, A^P, &c. 

 We see that the m*'' element of every arrangement in G, 

 by the rule of Art. 3, is placed in the same vertical row. 



We shall call GP the derangement of G by P. It is 

 known, and easily proved, that the derangement GP of G 

 by P is identical with the derangement of G by A^P, A^ 

 being any substitution of G. 



Let Q be an arrangement of the N elements which is 



