论文标题
平均角色程度和解决性的变化
Variations on average character degrees and solvability
论文作者
论文摘要
令$ g $为有限的组，$ \ bbb {f} $是字段$ \ mathbb {q}，\ mathbb {r} $或$ \ mathbb {c} $，而$ n $是非物质的普通子群为$ g $。令$ {\ rm acd} _ {\ bbb {f}}^{*}（g）$和$ {\ rm acd} _ {\ bbb {f}，甚至是所有非linear $ \ bbb f $ \ bbb $ f $ g $ g $ g $ g $ g $ g $ g $ g $ g $ f $ g $ f $ g $ f $ g $ f $ g $ f $ g $ f $ g $ f $ g $ f $ g $ f $ g $ f $ f $ g $ $ g $的f $不可减至的字符，其内核分别不包含$ n $。我们假设一个空套的平均值为$ 0 $，以方便起见。在本文中，我们证明，如果$ {\ rm acd}^*_ {\ mathbb {q}}}}（g）<9/2 $或$ 0 <{\ rm acd} _ {\ mathbb {q} _ {\ mathbb {q}，cank v}（g | n）<4 $，然后$ g $ solvable。此外，设置$ \ bbb {f} \ in \ {\ bbb {r}，\ bbb {c} \} $，我们通过假设$ {\ rm acd} _ {\ rm acd} _ {\ bb {\ bb {\ bb {f}}}}}}^{$ 0 <29 <29 <29/8 acd} _ {\ bbb {f}，均匀}（g | n）<7/2 $，我们得出结论$ n $的可溶性时，当$ 0 <{\ rm acd} _ {\ rm acd} _ {\ bbb {f}，甚至}（g | n）<18/5 $。将$ n $替换为$ g $ in $ {\ rm acd} _ {\ bbb {f}，偶数}（g | n）$，为我们提供了Moreto和Nguyen结果的扩展形式。举例说明所有边界都很尖锐。
Let $G$ be a finite group, $\Bbb{F}$ be one of the fields $\mathbb{Q},\mathbb{R}$ or $\mathbb{C}$, and $N$ be a non-trivial normal subgroup of $G$. Let ${\rm acd}_{\Bbb{F}}^{*}(G)$ and ${\rm acd}_{\Bbb{F},even}(G|N)$ be the average degree of all non-linear $\Bbb F$-valued irreducible characters of $G$ and of even degree $\Bbb F$-valued irreducible characters of $G$ whose kernels do not contain $N$, respectively. We assume the average of an empty set is $0$ for more convenience. In this paper we prove that if ${\rm acd}^*_{\mathbb{Q}}(G)< 9/2$ or $0<{\rm acd}_{\mathbb{Q},even}(G|N)<4$, then $G$ is solvable. Moreover, setting $\Bbb{F} \in \{\Bbb{R},\Bbb{C}\}$, we obtain the solvability of $G$ by assuming ${\rm acd}_{\Bbb{F}}^{*}(G)<29/8$ or $0<{\rm acd}_{\Bbb{F},even}(G|N)<7/2$, and we conclude the solvability of $N$ when $0<{\rm acd}_{\Bbb{F},even}(G|N)<18/5$. Replacing $N$ by $G$ in ${\rm acd}_{\Bbb{F},even}(G|N)$ gives us an extended form of a result by Moreto and Nguyen. Examples are given to show that all the bounds are sharp.
