COMPLEX ALGEBRAS 
181 
§ 103! 
(23) 
(p=i, 2, ... .) 
form a complete set of linearly independent elements of 
N of character 
(24) 
whence every element of that character is a linear func 
tion of the elements (23). By (23), 
whence P p is of character (22). Since N is invariant in 
A, k p belongs to N. We shall prove that the P p , with 
i, j, a, f3 fixed, form a complete set of linearly independent 
elements of N of character (22). First, if they were 
dependent, 'Lc p P p = o for complex numbers c p not all 
zero, we multiply by e\ a on the left and by on the right 
and get 
whence each c p = o, contrary to hypothesis. Hence the 
number of elements in a complete set of character (22) 
is not less than the number in a complete set of char 
acter (24). To prove the reverse, note that if a set of 
P p are linearly independent, the corresponding elements 
(23) will be linearly independent, since we saw how to 
deduce P p from (23) by multiplying by e\ x on the left 
and by e{p on the right.