{"title":"Character Table Groups and Extracted Simple and Cyclic Polygroups","authors":"Sara Sekhavatizadeh, M. M. Zahedi, A. Iranmanesh","doi":"10.22342/JIMS.26.1.742.22-36","DOIUrl":null,"url":null,"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.","PeriodicalId":42206,"journal":{"name":"Journal of the Indonesian Mathematical Society","volume":"26 1","pages":"22-36"},"PeriodicalIF":0.3000,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the Indonesian Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.22342/JIMS.26.1.742.22-36","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.