{"title":"在有限幂零无限群上","authors":"E. Detomi, M. Morigi","doi":"10.22108/IJGT.2019.119581.1577","DOIUrl":null,"url":null,"abstract":"Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite group where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements for any $x in G$. We prove that then $gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements, for any $xin G$.","PeriodicalId":43007,"journal":{"name":"International Journal of Group Theory","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On finite-by-nilpotent profinite groups\",\"authors\":\"E. Detomi, M. Morigi\",\"doi\":\"10.22108/IJGT.2019.119581.1577\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite group where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements for any $x in G$. We prove that then $gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements, for any $xin G$.\",\"PeriodicalId\":43007,\"journal\":{\"name\":\"International Journal of Group Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2020-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Journal of Group Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.22108/IJGT.2019.119581.1577\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Group Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.22108/IJGT.2019.119581.1577","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite group where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements for any $x in G$. We prove that then $gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$ elements, for any $xin G$.
期刊介绍:
International Journal of Group Theory (IJGT) is an international mathematical journal founded in 2011. IJGT carries original research articles in the field of group theory, a branch of algebra. IJGT aims to reflect the latest developments in group theory and promote international academic exchanges.