{"title":"Some Properties of Normal Subgroups Determined from Character Tables","authors":"Z. Akhlaghi, M. J. Felipe, M. K. Jean-Philippe","doi":"10.1007/s40840-024-01684-6","DOIUrl":null,"url":null,"abstract":"<p><i>G</i>-character tables of a finite group <i>G</i> were defined in Felipe et al. (Quaest Math, 2022. https://doi.org/10.2989/16073606/16073606.2022.2040633). These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of <i>G</i>. We analyze certain structural properties of normal subgroups which can be determined using their <i>G</i>-character tables. For instance, we prove an extension of the Thompson’s theorem from minimal <i>G</i>-invariant characters of a normal subgroup. We also obtain a variation of Taketa’s theorem for hypercentral normal subgroups considering their minimal <i>G</i>-invariant characters. This generalization allows us to introduce a new class of nilpotent groups, the class of <i>nMI</i>-groups, whose members verify that its nilpotency class is bounded by the number of irreducible character degrees of the group.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40840-024-01684-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
G-character tables of a finite group G were defined in Felipe et al. (Quaest Math, 2022. https://doi.org/10.2989/16073606/16073606.2022.2040633). These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of G. We analyze certain structural properties of normal subgroups which can be determined using their G-character tables. For instance, we prove an extension of the Thompson’s theorem from minimal G-invariant characters of a normal subgroup. We also obtain a variation of Taketa’s theorem for hypercentral normal subgroups considering their minimal G-invariant characters. This generalization allows us to introduce a new class of nilpotent groups, the class of nMI-groups, whose members verify that its nilpotency class is bounded by the number of irreducible character degrees of the group.
费利佩等人（Quaest Math, 2022. https://doi.org/10.2989/16073606/16073606.2022.2040633）定义了有限群 G 的 G 字符表。我们分析了可以通过 G 字符表确定的正则子群的某些结构性质。例如，我们证明了汤普森（Thompson）定理从正则子群的最小 G 不变字符出发的扩展。考虑到超中心正则子群的最小 G 不变字符，我们还得到了塔克塔定理的变体。通过这一概括，我们引入了一类新的无穷群，即 nMI 群，其成员验证了其无穷类是以群的不可还原特征度数为界的。
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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