论文标题
根据有限组的消失元素的顺序
On the orders of vanishing elements of finite groups
论文作者
论文摘要
令$ g $为有限的组，$ p $为素数。令$ \ mathrm {vo}（g）$表示消失元素的顺序集，$ \ mathrm {vo} _ {p}（g）$是$ \ mathrm {vo}（vo}（g）的子集$ \ mathrm {vo}（g）的$由这些消失元素的顺序组成，不可除以$ p $。 Dolfi，Pacifi，Sanus和Spiga证明，如果$ a $不是$ p $ - \ in \ mathrm {vo}（g）$的所有$ a \，则$ g $具有普通的Sylow $ P $ -Subgroup。在另一篇文章中，同一位作者还表明，如果如果$ \ mathrm {vo} _ {p'}（g）= \ emptyset $，则$ g $具有正常的nilpotent $ p $ - complement。这些结果是众所周知的ITO-Michler和Thompson定理的变化。在本文中，我们研究可解决的组，以便$ | \ m atrm {vo} _ {p}（g）| = 1 $，并表明$ p'$是亚正常。这类似于以撒，莫雷托，纳瓦罗和Tiep的工作，他们认为只有一个字符学位的小组可除以$ p $。我们还研究某些有限群$ g $，以便$ | \ m atrm {vo} _ {p'}（g）| = 1 $，我们证明$ g $具有普通的亚组$ l $，因此$ g/l $ a普通$ p $ - $ p $ - $ l $具有正常的$ p $ complent。这类似于Giannelli，Rizo和Schaeffer Fry最近在角色学位上的作品，并获得了几个$ p'$ - 角色学位。 Bubboloni，Dolfi和Spiga研究了有限组，因此对于某些整数$ M \ GEQSLANT 1 $，每个消失的元素都是订单$ p^{m} $。作为概括，我们调查了一个组，以使某些整数$ m \ geqslant 0 $，$ \ gcd（a，b）= p^{m} $，对于所有$ a，b \ in \ mathrm {vo}（g）$。我们还研究了可解决的有限群体，其不可约束的字符仅在主要功率秩序的要素上消失。
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, Moréto, 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.
