On the orders of vanishing elements of finite groups

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