Vanishing Elements of Prime Power Order

An element $x$ in a finite group $G$ is said to be $\mathit{\text{vanishing}}$ if some (complex) irreducible character of $G$ takes value $0$ at $x$. In this article, we prove that every non-abelian finite simple group, except ${\mathrm{S}\mathrm{L}}_2(4)$ and ${\mathrm{S}\mathrm{L}}_2(8)$, contains a vanishing element $\mathit{\text{of}}\mathit{\text{prime}}\mathit{\text{power}}\mathit{\text{order}}$ whose conjugacy class size is divisible by three distinct primes. We use this result to obtain the following generalization of a result of Robati ($2021$): If $G$ is a non-solvable finite group in which, the conjugacy class size of all the vanishing elements of prime power order has at most two distinct prime divisors, then $G\mathrm{/}\mathrm{S}\mathrm{o}\mathrm{l}(G)$ is a direct product of mutually isomorphic simple groups among ${\mathrm{S}\mathrm{L}}_2(4)$ and ${\mathrm{S}\mathrm{L}}_2(8)$. ($\mathrm{S}\mathrm{o}\mathrm{l}(G)$ is the largest normal solvable subgroup of $G$.)