Trace of inverse is complex conjugate of trace

Statement
Suppose $$g$$ is an invertible linear transformation of finite order acting on a finite-dimensional $$\mathbb{C}$$-vector space. Then, the trace of $$g^{-1}$$ is the complex conjugate of the trace of $$g$$.

Facts used

 * uses::Element of finite order is semisimple and eigenvalues are roots of unity

Proof
Using the above fact about eigenvalues, we see that $$g$$ is diagonalizable and all its complex eigenvalues are roots of unity. In particular, every eigenvalue is on the unit circle and its inverse equals its complex conjugate.

The eigenvalues of $$g^{-1}$$ are, element-wise, the inverses of the eigenvalues of $$g$$. Hence, the trace of $$g^{-1}$$ is the sum of the inverses of the eigenvalues of $$g$$, and hence equal to the sum of the complex conjugates of the eigenvalues of $$g$$. Since complex conjugation is additive, the trace of $$g^{-1}$$ is the complex conjugate of the trace of $$g$$.