CHAPTER XIV. GROUP OF THE EQUATION FOR THE 27 STR. LINES etc. 303 

 The latter only serves to define the operator I in terms of A and Bi 



The resulting expressions for S i} S i} 8? are seen to be commutative 

 and of period 2, so that relations a) follow from 282). 



282. The group LF(2, 7) of order 168 is defined by relations 

 a), D), I 2 = I, together with the following N = 4 relations 



8 6 T8^T8 6 T=I f S B TS 5 TS 2 TS 6 TS 2 T= I, 



Applying a), D) and T 2 = 1, the second and third relations become 



SiZS^TSs - S l TS l T8 l - S s = TS 5 1S 5 . 

 The first relation may be written S 9 T8 t TS S^TS^TS^ - 8^1 = 1 or 



The fourth relation becomes an identity if we replace S$TS 5 I by 

 ZS 5 TSTS 5 as derived from the first relation. Hence the 6r 168 may 

 l)e generated ~by 8 L and T subject only to the generational relations^) 



283) T* = I, 81 = I, (^T) 3 = J, (8}T)* = I. 



Corollary. The group LF(3, 2) of order 168 is isomorphic 

 with LF(2, 7). In fact, the relations 283) are satisfied by the sub- 

 stitutions 



CHAPTER XIV. 



GROUP OF THE EQUATION FOR THE 27 STRAIGHT LINES 

 ON A GENERAL SURFACE OF THE THIRD ORDER. 8 ) 



283. A general cubic surface contains 27 straight lines such that 3 ) 

 1. Any one of the lines A meets ten others which intersect 

 two by two, forming with A five triangles. The total number of 

 such triangles on the cubic surface is 5 27/3 = 45. 



1) Dyck, Math. Ann., vol. 20, p. 41; Burnside, The Theory of Groups, p. 305. 



2) Compare Jordan, Traite", pp. 316 329, 365 369; Dickson, Comptes 

 Rendus, vol. 128, pp. 873 875. 



3) Steiner, Crelle, vol. 53. 



