I was wondering if the following is true:
Let $\chi$ and $\chi'$ be two irreducible characters over $\mathbb{C}$ of a finite group with same degree.
Suppose that $\chi(g) = 0 \implies \chi'(g) = 0$.
Is true that $\chi'(g) = 0 \implies \chi(g) = 0$?
In other words, can the zero set of a character be a strict subset of the zero set of another character of the same degree?
Any counter-example or help proving this is appreciated.
Best Answer
Check out the dihedral group $D_{12}$.