SUMMARY OF THE KNOWN SYSTEMS OF SIMPLE GROUPS. 309 



289. Theorem. 1 ) -- The simple groups A(^m,yf) and 



F0(2m + l,p), P>2, 



of equal order are not isomorphic if m > 2. 



.The proof consists in showing that the orthogonal group contains 

 a greater number of sets of conjugate operators of period two than 

 the Abelian group. By 122, A(2m, p n ), p>2, has exactly 



(m + 2) or y (m -j- 1) distinct sets of conjugate operators of period 



two according as m is even or odd. But F0(2m+l,p n ) contains 

 the following m distinct substitutions of period two, 



C^C^j C-f)^C^C^ . . ., C^CjC/jt/a . . . C^m iCsm? 



having the respective characteristic determinants, 



(1 + KY (1 - X) 2 - 1 , (1 + K)* (1 - ) 2m - 3 , ., (1 + E)* m (1 - #) 



By 102, no two of these m substitutions are conjugate under linear 

 transformation. 



For m = 1 or for m = 2, the corresponding groups are iso- 

 morphic ( 288). 



290. The following table gives the 53 known simple groups of 

 composite order less than one million. The alternating group on 



n letters is designated by its order ^n\ The isomorphisms indicated 

 in 288 are not given in the table. 



LF(2, 2 2 ) ~ y 5! 



9 2) 



6072 



7800 



7920 



9828 



12180 



14880 



20160 



20160 



25308 



25920 



32736 



34440 



39732 



LF(2, 23) 

 LF(2, 5 2 ) 



Group on 9 letters 2 ) 

 LF(2, 3 3 ) 

 LF(2, 29) 

 LF(2, 31) 



>,2 2 ) 

 LF(2, 37) 



:, 3) - #0(4, 2 2 ) 

 ,2 5 ) 



, 43) 



in 238 



1) The existence of two non- isomorphic groups of order 81 was noted 



2) Cole, Quart. Journ. of Math., vol. 27, p. 48, foot-note. 



