Exercise 2.13, I. Martin Isaacs' Character Theory
Solution 1:
Since $\chi$ is nonlinear, $G'$ is not a subset of the kernel, so $\ker(\chi) \cap G' = 1$ since $|G'|=p$ is so small. Hence $[g,h]=1$ and your argument is complete.
Exercise 2.13, I. Martin Isaacs' Character Theory
Solution 1:
Since $\chi$ is nonlinear, $G'$ is not a subset of the kernel, so $\ker(\chi) \cap G' = 1$ since $|G'|=p$ is so small. Hence $[g,h]=1$ and your argument is complete.