Let be a finite subgroup of this is the group of the invertible matrices over ). If then prove that .

**Solution**

Let us suppose that and . We note that for every the depiction such that is and onto. Thus:

Thus the matrix is idempotent. thus its trace equals to its class. (since we are over which is a field of zero characteristic.) Hence

This implies that hence .

The exercise can also be found at mathematica.gr