A GENERALIZATION OF THE ABELIAN LINEAR GROUP. 



113 



In virtue of the relations 95), the conditions 89) all reduce to 

 identities. In fact, in each relation 89), at least two of the J'B are 

 distinct, say j ={= j 2 , and therefore all minors formed from the first, 

 and second columns vanish in virtue of 95). 



A substitution S belongs to the group G(m, q, p n \ q > 2, if and 

 only if its coefficients satisfy the conditions 88) and 95). 



127. Theorem. - - Every substitution S leaving <t> invariant can 

 be derived from the totality of linear substitutions of determinant unity 

 on q indices 



Xik (J - V- > 2), 



together with the linear substitutions, each on 2q indices, 



... (XigXjg) (i, j-1, ..., l). 



We can evidently derive from these generators a substitution T 

 which belongs to Gr(m, q, p n ) and replaces an arbitrary index Xti by 

 any particular index as x n . We may therefore suppose that in the 

 product S' = TS, S being defined by 87), the coefficient ag =|= 0. 

 If then we set 



}!-= CJ*g (j - 2, . . , m; ft - 1, . . ., q) 



it follows from 95), for i 1, r 1, / = 1, # 1, j > 1, that 



96) J* = Qi J* ( j = 2, . . ., m\ fc, s = 1, . . ., g). 



Substituting these values in the relation 91) for j = 1, we find 



t n . . . 



2-1 



2-1 

 = 1. 



It follows that 



aft 



Hence the following substitution is of determinant D =)= 0. 



DlCKSON, Linear Groups. 



