Character Table Groups and Extracted Simple and Cyclic Polygroups

Sara Sekhavatizadeh, M. M. Zahedi, A. Iranmanesh
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Abstract

Let ${G}$ be a finite group and $\hat{G}$ be the set of all irreducible complex characters of $G.$ In this paper, we consider $ $ as a polygroup, where for each $\chi _{i} ,\chi_{j}\in \hat{G}$ the product $\chi _{i} * \chi_{j}$ is the set of those irreducible constituents which appear in the element wise product $\chi_{i} \chi_{j}.$ We call that $\hat{G}$ simple if it has no proper normal subpolygroup and show that if $\hat{G}$ is a single power cyclic polygroup, then $\hat{G}$ is a simple polygroup and hence $\hat{S}_{n}$ and $\hat{A}_{n}$ are simple polygroups. Also, we prove that if $G$ is a non-abelian simple group, then $\hat{G}$ is a single power cyclic polygroup. Moreover, we classify $\hat{D}_{2n}$ for all $n.$ Also, we prove that $\hat{T}_{4n}$ and $\hat{U}_{6n}$ are cyclic polygroups with finite period.
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字符表群和提取的简单和循环多群
设${G}$是一个有限群,$\hat{G}$是$G在本文中,我们将$$视为一个多群,其中对于每一个$\chi_{i},\chi_{j}\In\hat{G}$,乘积$\chi_{i}*\chi_{j}$是出现在逐元素乘积$\chi{i}\chi}j}中的那些不可约成分的集合。$我们称之为$\hat{G}$简单,如果它没有适当的正规亚群,并证明如果$\hat{G}$\hat{G}$是单幂循环多群,那么$\hat}G$是一个简单多群,因此$\hat{S}_{n} $和$\hat{A}_{n} $是简单的多边形。此外,我们还证明了如果$G$是非阿贝尔单群,那么$\hat{G}$是单幂循环多群。此外,我们对$\hat进行了分类{D}_{2n}$用于所有$n.$此外,我们证明$\hat{T}_{4n}$和$\hat{U}_{6n}$是具有有限周期的循环多群。
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CiteScore
0.70
自引率
33.30%
发文量
20
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