{"title":"字符表群和提取的简单和循环多群","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":"{\"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}","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}
Character Table Groups and Extracted Simple and Cyclic Polygroups
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.