When you browse the character tables of the small finite groups (for example here), you can observe that every zero entry corresponds to the value of an irreducible character $\chi$ on a non-central element $g$ such that the degree $\deg(\chi)$ of $\chi$ and the order $|C_G(g)|$ of the centralizer of $g$ in $G$ are not coprime (i.e. $\gcd(\deg(\chi) , |C_G(g)| ) \neq 1$).

**Question 1**: Is it true in general?

The reciprocal is false, one counter-example is given by $S_4$ which admits an irreducible character $\chi$ and a non-central element $g$ with $\deg(\chi) = 2$, $|C_G(g)| = 8$ but $\chi(g) = 2$. Moreover, the vanishing of $\chi(g)$ for $\chi$ irreducible ang $g$ non-central is not completely determined by $\deg(\chi)$ and $|C_G(g)|$ because for $G = M_{11}$ there is $\chi$ irreducible, $g_1, g_2$ non-central with $\deg(\chi) = 10$, $|C_G(g_1)| = |C_G(g_2)| = 8$, but $\chi(g_1) = 0$ and $\chi(g_2) = 2$.

**Question 2**: Why for $\chi$ irreducible, $g$ non-central and $\gcd(\deg(\chi) , |C_G(g)| ) \neq 1$ then "**often**" $\chi(g)$ vanishes? Is it always true in some specific cases, for example when $\deg(\chi) = |C_G(g)| $?

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