论文标题
在一个消失的班级大小的可解决组上
On solvable groups with one vanishing class size
论文作者
论文摘要
令$ g $为有限的组，让CS $（g）$成为$ g $的共轭类尺寸。回想一下，如果存在$ g $的$ g $ g $ $ g $的$ g $ a \ emph {nashing element}，则$ g $的不可约性特征，$ g $的值$ 0 $上的$ g $上的值$ 0 $，我们考虑的一个特定子集CS $（g）$的一个特定子集，即，该元素是其元素是$ G $的conjugacy class sents nes nest vcs $（g）$ g $ g $ g $ g $ g $ g $。由\ cite {blp}的结果激励，我们描述了有限组$ g $的类别，以便VCS $（g）$由单个元素\ emph {在假设$ g $中$ g $ supersolvable或$ g $具有正常的sylow $ 2 $ -subgroup}（特别是odd Odd Ord ofd odd cord）的假设。作为一种特殊情况，我们还获得了有限组的表征，该组具有单个消失的共轭类大小\ emph {the是素数或无方形}。
Let $G$ be a finite group, and let cs$(G)$ be the set of conjugacy class sizes of $G$. Recalling that an element $g$ of $G$ is called a \emph{vanishing element} if there exists an irreducible character of $G$ taking the value $0$ on $g$, we consider one particular subset of cs$(G)$, namely, the set vcs$(G)$ whose elements are the conjugacy class sizes of the vanishing elements of $G$. Motivated by the results in \cite{BLP}, we describe the class of the finite groups $G$ such that vcs$(G)$ consists of a single element \emph{under the assumption that $G$ is supersolvable or $G$ has a normal Sylow $2$-subgroup} (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size \emph{which is either a prime power or square-free}.
