306 CHAPTER XIV. GEOUP OF THE EQUATION FOR THE 27 STR. LINES etc. 



285. Theorem. The group G- of the equation for the 27 lines 

 on a general cubic surface is of order 51840 and has a subgroup of 

 index 2 holoedrically isomorphic with the abstract group 0. 



The group G is formed of the substitutions on the 27 elements E 

 which permute the 45 triangles. These substitutions can replace R Q 

 by at most 27 elements. Those leaving R Q fixed can replace E L by 

 no element other than the ten lying with R in some triangle; 

 namely, _R 1; R 2 , RUO, ftio (i = 1, 2, 3, 4). The substitutions leaving 

 R Q and R t fixed and consequently the triangle R R 1 R 2 cannot alter 

 R 2 and must replace -R 130 by one of the 8 elements 



which enter the four remaining triangles containing R . The sub- 

 stitutions leaving R , R lf J? 130 fixed cannot alter _R 2 or R 2ZQ , and 

 must permute amongst themselves the triangles which contain j^ and 

 likewise the triangles which contain -R 130 . Hence they must permute 

 the pairs R ni , .R 211 ; -R m , -R 221 ; -R 131 , -R 232 5 -^ui; ^24i5 an d likewise 

 permute the pairs R ln , ^ 232 ; .R 212 , .R 231 ; R 122 , R U1 ; J? 121; JR 142 . Hence 

 the elements J? m , -^121? ^231? -^141 common to the two sets must be 

 permuted amongst themselves, which can be done in at most 24 ways. 

 Finally, a substitution of 6r which leaves fixed R Q , JR 1; _R 2 , -R 130 , -R 2307 



^m, -Biau ^231 and -Bui must not alter -Rgu; ^221; Asi; ^i, ^232. 

 JR 212 , .R 122 and R 142 and therefore must leave fixed the third element 



in each of the triangles 2^u^ 12 ^ 10 , ^241-^232 -^220 , ^221^122^110; 



-^210-^121-^222^ -^230-^131-^112? ^230-^22 1-^2 42; -^131 -^142 -^120 ; -^ 22-^531-^*40? 



and ^m-Ruo^m- Such a substitution therefore leaves fixed every 

 element and is therefore the identity. The order of 6r is therefore 

 at most 27 10 - 8 - 24 = 51840. 



But G contains the subgroup [0] of order 25920 whose sub- 

 stitutions permute the 45 triangles evenly. Also G contains 



which gives rise to the following odd substitution on the triangles: 



R t Rut Rut) 



w ti> RutRn t +1-^24 1 i) 



(Rs2tRs3 f+1^4 t I, Rs2 tlRsltRsS t+l) 



containing 3 + 3 + 3 + 6 = 15 transpositions. The order of G is 

 therefore at least 2 - 25920. The order is consequently 51840. 



286. Certain subgroups of the abstract group of order 25920 

 appear at once by considering the various isomorphic linear groups. 



