**Theorem** (Maschke, 1899) Let be a field and let be a finite group of order Let Then is semiprimitive if and only if

*Proof*. Let where Suppose first that and consider the algebra homomorphism defined by for all Define by for all Note that here is the trace of the matrix corresponding to the linear transformation with respect to the ordered basis We remark a few points about :

1) because is the identity map of

2) If then The reason is that for all and thus the diagonal entries of the matrix of are all zero and so

3) If is a nilpotent element of then because then for some and thus So is nilpotent and we know that the trace of a nilpotent matrix is zero.

Now let Since is finite dimensional over it is Artinian and hence is nilpotent. Thus is nilpotent and therefore, by 3), Let where Then

by 1) and 2). It follows that because So the coefficient of of every element in is zero. But for every the coefficient of of the element is and so we must have Hence and so

Conversely, suppose that and put Clearly for all and hence

Thus is a nilpotent ideal of and so Therefore because