On the orders of vanishing elements of finite groups

Let $G$ be a finite group and $p$ be a prime. Let $\mathrm{Vo}(G)$ denote the set of the orders of vanishing elements, $\mathrm{Vo}_{p} (G)$ be the subset of $\mathrm{Vo}(G)$ consisting of those orders of vanishing elements divisible by $p$ and $\mathrm{Vo}_{p'} (G)$ be the subset of $\mathrm{Vo}(G)$ consisting of those orders of vanishing elements not divisible by $p$. Dolfi, Pacifi, Sanus and Spiga proved that if $a$ is not a $p$-power for all $a\in \mathrm{Vo}(G)$, then $G$ has a normal Sylow $p$-subgroup. In another article, the same authors also show that if if $\mathrm{Vo}_{p'}(G) =\emptyset$, then $G$ has a normal nilpotent $p$-complement. These results are variations of the well known Ito-Michler and Thompson theorems. In this article we study solvable groups such that $|\mathrm{Vo}_{p}(G)| = 1$ and show that $P'$ is subnormal. This is analogous to the work of Isaacs, Mor\'eto, Navarro and Tiep where they considered groups with just one character degree divisible by $p$. We also study certain finite groups $G$ such that $|\mathrm{Vo}_{p'}(G)| = 1$ and we prove that $G$ has a normal subgroup $L$ such that $G/L$ a normal $p$-complement and $L$ has a normal $p$-complement. This is analogous to the recent work of Giannelli, Rizo and Schaeffer Fry on character degrees with a few $p'$-character degrees. Bubboloni, Dolfi and Spiga studied finite groups such that every vanishing element is of order $p^{m}$ for some integer $m\geqslant 1$. As a generalization, we investigate groups such that $\gcd(a,b)=p^{m}$ for some integer $m \geqslant 0$, for all $a,b\in \mathrm{Vo}(G)$. We also study finite solvable groups whose irreducible characters vanish only on elements of prime power order.