CHAPTER Xin. AUXILIARY THEOREMS ON ABSTRACT GROUPS, etc. 287 



the tetrahedral 6r 12 . For s = 11, 3 2 , 7, 5, the subgroups of maximum 

 order are 6r 60 , 6r 60 , 6r 24 , 6r 12 respectively, the index under GM^ being 

 11, 6, 7, 5 and hence < s + 1. For s = 2, 3 the G- M(t ) is a dihedron 6r 6 , 

 a tetrahedron 6r 12 , respectively, and has a subgroup of maximum 

 order 6r 3 , 6r 4 respectively. 



263. A simple group can be represented as a transitive sub- 

 stitution-group on N letters if, and only if, it contains a complete 

 system of N conjugate subgroups. 1 ) For s > 3, G M () is simple ( 261). 

 Hence G M ( a ) can be represented as a transitive group on < s -f- 1 letters 

 only when s = 5, 7, 3 2 , 11. For s = 2, 3 it can be represented as a 

 transitive group on 3, 4 letters respectively, but on no fewer, being 

 of order 6r 6 , 6r 12 . If a simple group be represented as an intransitive 

 substitution -group on D letters, D must equal the sum of the degrees 

 of two or more transitive representations; for GM(S) we have always 

 _D > s 4- 1. Hence the linear fractional group G-JJ( S ) may be represented 

 as a substitution -group on s -f 1 letters but on no fewer number except 

 when s 5, 7, 9, 11, for which the minimum number of letters is 5, 7, 

 6, 11 respectively. 



CHAPTER XHL 



AUXILIARY THEOREMS ON ABSTRACT GROUPS. ABSTRACT 

 FORMS OF VARIOUS LINEAR GROUPS. 2 ) 



264. Theorem. The symmetric substitution- group on k letters 

 is holoedrically isomorphic with the abstract group G(k) generated by 

 the operators S ly B 2 , . . ., JRt_i with the generational relations 



261) 5f = ^ = ... = jg|_ 1 = J, 



262) BtBj^BjB; (i = l, 2, . . ., k- 3; j = i + 2, i + 3, . . ., fc-1), 



263) S j S j+1 S j = B j+l S j S j+l O' = l,2, ...,&-2). 



The symmetric group 6r*? on the letters Z 1? ? 2 , . . ., Z* may be 

 generated by the transpositions 



which satisfy the relations 261), 262), 263) prescribed for the 

 generators B d of the abstract group 6r(fc) and conceivably also other 



1) For a proof of this theorem due to Dyck see Burnside, The Theory of 

 Groups, 123. 



2) The theorems of 264, 265 are due to Professor Moore , Proceed. Land. 

 Math. Soc., vol. XXVIII, pp. 357 366. The proofs given in 264, 266 are due 

 to the author; for that in 264 see Proceed. Lond. Math. Soc., vol. XXXI, 

 351353; for that in 266 see Math. Ann., vol t 64, pp. 564 569. 



