{"title":"元素的子群、次正规和相对序的乘积","authors":"Luca Sabatini","doi":"10.26493/1855-3974.2975.1b2","DOIUrl":null,"url":null,"abstract":"Let $G$ be a group. We give an explicit description of the set of elements $x \\in G$ such that $x^{|G:H|} \\in H$ for every subgroup of finite index $H \\leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?","PeriodicalId":49239,"journal":{"name":"Ars Mathematica Contemporanea","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Products of subgroups, subnormality, and relative orders of elements\",\"authors\":\"Luca Sabatini\",\"doi\":\"10.26493/1855-3974.2975.1b2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $G$ be a group. We give an explicit description of the set of elements $x \\\\in G$ such that $x^{|G:H|} \\\\in H$ for every subgroup of finite index $H \\\\leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?\",\"PeriodicalId\":49239,\"journal\":{\"name\":\"Ars Mathematica Contemporanea\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-10-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Ars Mathematica Contemporanea\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.26493/1855-3974.2975.1b2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Mathematica Contemporanea","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.26493/1855-3974.2975.1b2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Products of subgroups, subnormality, and relative orders of elements
Let $G$ be a group. We give an explicit description of the set of elements $x \in G$ such that $x^{|G:H|} \in H$ for every subgroup of finite index $H \leqslant G$. This is related to the following problem: given two subgroups $H$ and $K$, with $H$ of finite index, when does $|HK:H|$ divide $|G:H|$?
期刊介绍：
Ars mathematica contemporanea will publish high-quality articles in contemporary mathematics that arise from the discrete and concrete mathematics paradigm. It will favor themes that combine at least two different fields of mathematics. In particular, we welcome papers intersecting discrete mathematics with other branches of mathematics, such as algebra, geometry, topology, theoretical computer science, and combinatorics. The name of the journal was chosen carefully. Symmetry is certainly a theme that is quite welcome to the journal, as it is through symmetry that mathematics comes closest to art.
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