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



This proves that the RjyN substitutions of the form 



{pj + c) (mod. N) ip,,"^!) form a model group. 



The Rjv-i derived groups given by R^v-i values of j9> i, 



VG=piS{i + c), 



are the R^y- i derangements 



GV=S{i + c)pi; 



for pi{i + c)—p{i + c)=pi+pc=pi + c' and {i + c)pi=pi + c; 



that is, every substitution in PG is a substitution in GP. 



The group of EjyN substitutions has none of the order 



N except those of G=:S(i + c). For if any of the derived 



derangements added to G contained Q of the order N, 



there would be the N - i derived derangements 



QG a^G-.Q^'-^Gi (Art. 8); 



but we know that G has not more than N - 2 derived 



derangements, for Ry-i::j>N-2. 



We shall presently show how the groups equivalent to 



the simple modular group 



G' = S{pi + c), 



where p'^t has R^ values and c<N has N values, may 



be constructed, each of N • R^^ substitutions. 



16. Let q be any integer less than N and prime to it, 



and let it be a prime root of the-congruence 



x'' -1^0 (mod. N), 



where g'^Rjv? that is to say, such a root that we cannot 



have 



q^^^i (mod. N), 



Ti being less than r. 



We know that q^ is also a root of the congruence 



^'■-1^0 (mod. N). 

 We have proved that the products 



qiS{i + c), qHS{i + c)' -q'-HSii + c) 

 are derived derangements of the group 



G = S(i + c), 

 and they consequently form with G a group of Nr substi- 

 tutions of the form 



