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, Moreto, 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.

中文翻译：

---

关于有限群的消失元的阶

令 $ G$ 为有限群，$p$ 为素数。令 $\mathrm{Vo}(G) $ 表示消失元素的阶数集合，$\mathrm{Vo}_{p} (G)$ 是 $ \mathrm{Vo}(G) $ 的子集，包括那些可被 $p$ 和 $\mathrm{Vo}_{p'} (G) $ 整除的消失元素的顺序是 $ \mathrm{Vo}(G) $ 的子集，其中包含那些不可整除的消失元素的顺序由 $p$。Dolfi、Pacifi、Sanus 和 Spiga 证明，如果 $ a $ 不是所有 $ a\in \mathrm{Vo}(G)$ 的 $ p $-幂，则 $ G $ 具有正常的 Sylow $ p $-子群. 在另一篇文章中，同样的作者还表明，如果 $\mathrm{Vo}_{p'}(G) =\emptyset $，则 $G $ 具有正常的幂零 $p $-补码。这些结果是众所周知的 Ito-Michler 和 Thompson 定理的变体。在本文中，我们研究可解群，使得 $|\mathrm{Vo}_{p}(G)| = 1 $ 并表明 $ P' $ 不正常。这类似于 Isaacs、Moreto、Navarro 和 Tiep 的工作，他们认为只有一个字符度数的组可以被 $ p $ 整除。我们还研究了某些有限群 $G$ 使得 $|\mathrm{Vo}_{p'}(G)| = 1 $ 并且我们证明 $ G $ 有一个正规子群 $ L $ 使得 $ G/L $ 是一个正规的 $ p $-complement 并且 $ L $ 有一个正规的 $ p $-complement。这类似于 Giannelli、Rizo 和 Schaeffer Fry 最近关于具有几个 $p'$-character 度数的字符度数的工作。Bubboloni、Dolfi 和 Spiga 研究了有限群，使得每个消失元素对于某个整数 $ m\geqslant 1 $ 都是 $ p^{m} $ 阶。作为概括，我们调查了这样的组，$ \gcd(a, b)=p^{m} $ 对于某些整数 $ m \geqslant 0 $，对于所有 $ a,b\in \mathrm{Vo}(G) $。我们还研究了其不可约特征仅在素幂阶元素上消失的有限可解群。