Determinantal order of character of a group.

Question

The notion of determinantal order can be found in 'Character Theory of finite groups' by I Martin Isaacs.

If $\chi$ be a linear character of a finite group G, show that the order of $\chi$ in the group of linear characters of G is $\frac{|G|}{|\rm{ker}\chi|}$.

Answer 1

First isomorphism theorem:

$$\chi:G\to \Bbb C^*\;\;\text{homomorphism}\implies G/\ker\chi\cong\chi(G)\implies\ldots$$