THE ABELIAN LINEAR GROUP. 105 



To complete the proof of our theorem , we note that 



T 1 T T 1 T T W T T 



! 3= -ii,_i, J-2 -tl, 1 -L2,~ 1, ., -t* -^1, 1 ^2, 1 . -//H, 1 



have the respective characteristic determinants (with parameter K) 



Hence no two of them are conjugate under linear transformation. 



The most general substitution of SA(2m, p n ) commutative with T r 

 is seen to be A = A r A m r , where A r is an arbitrary special Abelian 

 substitution on the indices | f -, r?, : (i = 1, . . ., r) and A m - r an arbitrary 

 one on the indices /, rj t (i = r + 1, . . ., m). By 115 the number 

 of substitutions A r and A m r is respectively SA[2r, p n ] and 

 SA [2m 2r, p*]. Dividing SA[2m, p*} by the product of the 

 foregoing numbers, we obtain the number of substitutions of 

 SA(2m 9 p n ) conjugate with T f within the group. 



Operators of period 2 of A(2m,p*), 122123. 



122. By 119, we obtain the quotient -group A (2m, p n ) by 

 considering as identical S and ST= TS, where S is an arbitrary 

 substitution of SA(2m, p n ) and T is the self -conjugate substitution 

 Tj,_iT 2 ,_i . . . T m? _i. In particular, T r and T r T become identical 

 in the quotient -group. But the latter is conjugate with T OT _ r . 

 Furthermore, if s = m/2 or (m 1)/2 according as m is even or odd, 

 no two of the operators Ji, T*, . . ., T s are conjugate within the 

 quotient -group. The special Abelian substitutions of period 2 lead 

 therefore to just s distinct sets of conjugate operators of A(2m, p n \ 

 p > 2. To complete the study of the operators of period 2 of 

 A (2m, p n ), it remains to determine the conjugacy of the special 

 Abelian substitutions S for which S 2 =I. Being of period 4, such 

 an S is not conjugate to any T r . Moreover, no two of the cor- 

 responding operators of the quotient -group are conjugate, since that 

 would require one of the four relations 



= T r or TT r , A" l (8T)A^T r or TT r , 



A being Abelian. But any of these would require that 8 be conju- 

 gate with some T f . within the special Abelian group, whereas their 

 periods are different. Making use of the result of 123, we may 

 state the theorem: 



According as m is even or odd, the group A(2m, p n ), p > 2, has 

 exactly -- (m + 2) or (m + 1) distinct sets of conjugate operators of 

 period 2. 



