{"title":"The conjugacy diameters of non-abelian finite $ p $-groups with cyclic maximal subgroups","authors":"Fawaz Aseeri, J. Kaspczyk","doi":"10.3934/math.2024524","DOIUrl":null,"url":null,"abstract":"Let $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if $ G $ is the normal closure of $ S $ in $ G. $ In this case, any element of $ G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ g\\in G $ and $ S $ is a normally generating subset of $ G, $ then we write $ \\| g\\|_{S} $ for the length of a shortest word in $ \\mbox{Conj}_{G}(S^{\\pm 1}): = \\{h^{-1}sh | h\\in G, s\\in S \\, \\mbox{or} \\, s{^{-1}}\\in S \\} $ needed to express $ g. $ For any normally generating subset $ S $ of $ G, $ we write $ \\|G\\|_{S} = \\mbox{sup}\\{\\|g\\|_{S} \\, |\\, \\, g\\in G\\}. $ Moreover, we write $ \\Delta(G) $ for the supremum of all $ \\|G\\|_{S}, $ where $ S $ is a finite normally generating subset of $ G, $ and we call $ \\Delta(G) $ the conjugacy diameter of $ G. $ In this paper, we derive the conjugacy diameters of the semidihedral $ 2 $-groups, the generalized quaternion groups and the modular $ p $-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"125 6","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-01-17","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/10.3934/math.2024524","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
Let $ G $ be a group. A subset $ S $ of $ G $ is said to normally generate $ G $ if $ G $ is the normal closure of $ S $ in $ G. $ In this case, any element of $ G $ can be written as a product of conjugates of elements of $ S $ and their inverses. If $ g\in G $ and $ S $ is a normally generating subset of $ G, $ then we write $ \| g\|_{S} $ for the length of a shortest word in $ \mbox{Conj}_{G}(S^{\pm 1}): = \{h^{-1}sh | h\in G, s\in S \, \mbox{or} \, s{^{-1}}\in S \} $ needed to express $ g. $ For any normally generating subset $ S $ of $ G, $ we write $ \|G\|_{S} = \mbox{sup}\{\|g\|_{S} \, |\, \, g\in G\}. $ Moreover, we write $ \Delta(G) $ for the supremum of all $ \|G\|_{S}, $ where $ S $ is a finite normally generating subset of $ G, $ and we call $ \Delta(G) $ the conjugacy diameter of $ G. $ In this paper, we derive the conjugacy diameters of the semidihedral $ 2 $-groups, the generalized quaternion groups and the modular $ p $-groups. This is a natural step after the determination of the conjugacy diameters of dihedral groups.
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.