On the orders of vanishing elements of finite groups

Abstract

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.