标题： 关于字符码度与$p$-群的最小忠实拟排列表示次数之间的关系 显示英文标题

标题： On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of $p$-groups

摘要： 对于有限群$G$，我们用$c(G)$表示在复数域$\mathbb{C}$上，$G$通过拟置换矩阵的忠实表示的最小次数。 对于不可约特征标$\chi$的$G$，$\chi$的码度定义为$\cod(\chi) = |G/ \ker(\chi)|/ \chi(1)$。 在本文中，我们建立了$c(G)$与某个非交换$p$-群$G$的一些不可约特征标的余次数的$\mathbb{Q}_{\geq 0}$-和之间的等式。 我们还研究了非交换 $p$-群的各种类中 $c(G)$ 与不可约特征标余次数之间的关系，例如，中心循环的 $p$-群，极大类的 $p$-群，GVZ $p$-群和其他群。 显示英文摘要

摘要： For a finite group $G$, we denote by $c(G)$, the minimal degree of faithful representation of $G$ by quasi-permutation matrices over the complex field $\mathbb{C}$. For an irreducible character $\chi$ of $G$, the codegree of $\chi$ is defined as $\cod(\chi) = |G/ \ker(\chi)|/ \chi(1)$. In this article, we establish equality between $c(G)$ and a $\mathbb{Q}_{\geq 0}$-sum of codegrees of some irreducible characters of a non-abelian $p$-group $G$ of odd order. We also study the relation between $c(G)$ and irreducible character codegrees for various classes of non-abelian $p$-groups, such as, $p$-groups with cyclic center, maximal class $p$-groups, GVZ $p$-groups, and others.