Quotient by kernel of a character

Question

Let $G$ be a finite group, and $\chi$ a linear character of $G$.

I've read that the quotient $G/ker(\chi)$ is cyclic. Why is this true?

I also wonder about the quotient if $\chi$ is not linear.

Answer 1

If $k$ is a field, then every finite subgroup of $k^{\ast}$ is cyclic.

For representations $(\rho,V)$ of $G$ of degree $>1$, the quotient $G/\operatorname{Ker}\rho$ can be isomorphic to any given finite group. This is because the symmetric group on $n$ elements is isomorphic to a subgroup of $\operatorname{GL}_n$.