152 



CHAPTER VI. 



The corresponding substitution of the second compound is 



y. . (h> h 1, ., 2m 

 \ i < i 2 



In virtue of 139), [a] 2 transforms Z into 



A, .72 



Inversely, if [] 2 transforms Z into \iZ, the relations 139, follow. 



161. Since the Abelian group GA(2m,p n ) contains the substitution 

 T: iS--& (* = !,... ,2m), 



it is (by 154) holoedrically or hemiedrically isomorphic with its 

 second compound according as p = 2 or p > 2. 



If /S belong to the special Abelian group SA(2m, p n ), so that 

 ft = l, the corresponding substitution [] 2 of the second compound 

 will leave Z absolutely invariant. Since S then has determinant 

 unity ( 114), [] 2 will leave absolutely invariant the Pfaffian 

 [1 2 ... 2m] ( 156). If in SA(2m, p n ) we consider 8 and TS to 

 be identical, we obtain the quotient -group A(2m, p n ). The latter is 

 therefore simply isomorphic with the second compound of SA(2m, p n ). 

 Applying 119, we may state the theorem: 



Except for (2m, p n ) = (2, 2), (2, 3) and (4, 2) the second compound 

 of SA(2m,p'" : ) is a simple group which leaves absolutely invariant the 

 Pfaffian [1 2 ... 2m] and the linear function Z. 



162. For 2m = 4, p > 2, we introduce as new variables 



The general substitution [a] 2 of the second compound of SA(4, p n ) 

 takes the form, in which the unaltered index Z v does not appear 1 ), 

 V V V V V 



J- J-1Q -L 1 A J-OQ -J-OA 



y 



JL oo 



1) In 164 below, the second compound [a] 2 of an arbitrary quaternary 

 linear homogeneous substitution is written in matrix form. 



