THE ABELIAN LINEAR GROUP. 99 



We next prove that, for m > 2, J contains L^ i. Since, for p = 2, 



Z^-MtL^Mi, R^-MtMjNwMtfy, 

 it follows that J will contain the substitution 



-D = L[ t i L^ i 3, 1 -Bl, 2, 1 -Ra, 3, 1 ^3, 1, 1; 



the latter heing the product of an even number 24 of the sub- 

 stitutions 85). This product is seen to be 



D: 8-g,, tf = ifc+!i+ a +ls (i~l, 2, 3). 

 But D is transformed into i^i by the following Abelian substitution 

 of period two: 



Hence 7 contains .L^i and therefore also 



/, 1 -^1, 1 ' ^1, 1 ^/, 1 ^i, * /, 1 ^f, r = -/, t*> ^C i*i, i Zfi, i = -My, 



JV,, A i i,, ! iz, ! = N t , ^ lf T-/ JV;. A , ^ , = Nt,^. 



Hence, for p = 2, m > 2, J" is identical with 6r, so that 6r is simple. 

 For p = 2, m = 2, J" contains M M 2 as above, and therefore also 



MiM* T^ M t M 9 T^ ~ T^. 



Hence J contains every T^ a . But -Ri, 2, A transforms T^ a into 

 Ki, 2, 2(i + a) ^i, If w > 1, the 6rF[2*] contains a mark a neither 

 zero nor unity, so that 1 + a =j= 0, =)= 0. Hence, for n > 1, the 

 group J" contains JRi, 2,^(14- ) = ^i, 2,1? by proper choice of L It 

 therefore contains -A r i, 2, i- Having the products in pairs of the sub- 

 stitutions 85), J contains M{ and L,^. Thus J=G. 



The fact that the case m = 2, _p = 2, = 1 is exceptional is 

 shown in the following section. 



118. Theorem. - - The Abelian group SA(4, 2) on four indices 

 modulo 2 is holoedrically isomorphic with the symmetric group on six 

 letters. 1 ) 



By 264 of Chapter XIII, the symmetric group on 6 letters is 

 holoedrically isomorphic with the abstract group 6r 6! generated by 

 jBj, JBfc, -B 3 , -B 4 , B 5 subject to the generational relations 



j^2 = fil _ j^2 _ _g2 = _g2 = J 



J, 



B2 - L 



1) This theorem was first proved by Jordan by means of the groups of 

 Steiner, Traite', No. 335. The proof given in the text is due to the author, 

 Proc. Lond. Math. Soc., vol. 31, pp. 40 41. 



7* 



