I'm stuck in the c) part of the next problem
Let $\rho$ be a representation of the finite group $G$ over $\mathbb{C}$
a) Show that $\delta:g\rightarrow det(g\rho)\ (g\in G)$ is a linear character of $G$.
b) Prove that $G/Ker\ \delta$ is abelian.
c)Assume that $\delta (g)=-1$ for some $g\in G$. Show that $G$ has a normal subgroup of index 2.
So considering $\mathbb{R}$ instead of $\mathbb{C}$, we can define $N$ as the set of elements $g\in G$ with $\delta (g)>0$ and prove the result.
Considering $\mathbb{C}$, I don't know if it's true.
Any advice to prove or disprove it would be a great help.
As $G$ is a finite group, the image $\delta(G)$ is a finite subgroup $H$ of the multiplicative group of $\Bbb C$. As $-1\in H$, then $H$ has even order, and a finite abelian group of even order has a (normal) subgroup $K$ of index $2$. Then $\delta^{-1}(K)$ is an index two subgroup of $G$.