292 CHAPTER XIII. 



These results lead at once to the following correspondences: 

 (y s )I? 82 ~(265)(347), (y 2 | 4 y~(27)(3645), (8,WJ^~(24)(17), 



(y 8 )(| s i,)~(18)(34), (U*)^- (187) (234), 

 ; JS 32 ~(23)(45)(67)(18), (| 2 y~(18)(27)(35)(46). 



By simple transformations, we complete the proof of the 

 Theorem. - - The correspondences 270) give reciprocally 



(y 2 ) ~ (13)(27)(48)(56), (y s ) ~ (16)(27)(34)(58), 



(| ] y~(18)(27)(36)(45), 

 (y 3 ) ~ (18)(27)(S5)(46), (y 4 ) ~ (15)(27)(34)(68), 



(I 3 i 4 )~(14)(27)(38)(56), 

 5 12 ~ (12)(38)(47)(56), S sl ~ (17)(25)(34)(68), 



S 32 ~ (18)(23)(45)(67), 



14 ~ (18)(23)(46)(67), S 24 ~ (17)(26)(34)(58), 

 J? 48 ~(12)(37)(48)(56). 



By 100, tfAese relations enable us to pass from an arbitrary sub- 

 stitution of the linear group on 4 indices modulo 2 to the corresponding 

 even substitution on 8 letters. 



Abstract form of the simple group _FO(5, 3) 1 ), 270274 



270. By the notation of 194, F0(b, 3) denotes the group 0[(5, 3). 

 By 189 and 181, it is of order 25920 and is generated by the 

 substitutions 2 ) 



City, (!,&)(&&)> = "^i2S4 (^,j, *,? = !,... ,5). 



It has a commutative subgroup L 16 composed of the substitutions 



I) @1@2> ^1^3? ^1^4; ^1^5; ^2^3; ^2^4; Q@69 ^3^4? ^sQ? ^4^5> 



^iQC 8 C 4 , C&C&, C C B C,C 5 and C 2 C 3 4 (7 5 . The (&fe)(|*gO generate 

 a subgroup Z 60 of the even linear substitutions on | 1? . . ., | 5 . The 

 groups Z; 16 and Z^ 60 are commutative with each other and have only 

 the identity in common; hence they generate a subgroup ^ 60 of 

 FO (5, 3). We readily determine the abstract forms of these sub- 

 groups. By 265, we have the theorem: 



1) Taken from the author's papers, Comptes Rendus, vol. 128, pp. 873 875; 

 Proceed. Lond. Math. Soc., vol. 32, pp. 3 10. In the earlier paper, Proceed. 

 Land. Math. Soc., vol. 31, pp. 30 68, another set of generators was determined 

 by a more complicated analysis. 



2) For pn = 3, 0?>f is either the identity, C-C,, (i.-iy)^ or (1^)0,., the 

 first two alone being of the form Q.j. Here (1^) denotes the linear substitu- 

 tion |[- = |y, i^ = i f . They are to be compounded as linear substitutions; for 

 example, (iii 8 )(i 1 i 2 ) = (i 1 i s i 8 ). Also C i denotes the substitution changing the 

 sign of the index | 



