{"title":"自相似有限群的指数","authors":"Alex Carrazedo Dantas, Emerson de Melo","doi":"10.4171/ggd/754","DOIUrl":null,"url":null,"abstract":"Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\\{g\\in G | g^{p^n}=1\\}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.","PeriodicalId":55084,"journal":{"name":"Groups Geometry and Dynamics","volume":"329 ","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exponent of self-similar finite $p$-groups\",\"authors\":\"Alex Carrazedo Dantas, Emerson de Melo\",\"doi\":\"10.4171/ggd/754\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\\\\{g\\\\in G | g^{p^n}=1\\\\}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.\",\"PeriodicalId\":55084,\"journal\":{\"name\":\"Groups Geometry and Dynamics\",\"volume\":\"329 \",\"pages\":\"0\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-10-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Groups Geometry and Dynamics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4171/ggd/754\",\"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":"Groups Geometry and Dynamics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/ggd/754","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设$p$是一个素数，$G$是一个有限秩的亲$p$群，它在$p$任意根树上有忠实的、自相似的作用。证明了如果集合$\{g\在g | g^{p^n}=1\}$中是某$n$的非平凡子群，则$ g $是一个指数不超过$p^n$的有限$p$-群。这尤其适用于幂阿贝尔$p$-群。
Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G | g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.
期刊介绍：
Groups, Geometry, and Dynamics is devoted to publication of research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. The journal covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.
Topics covered include:
geometric group theory;
asymptotic group theory;
combinatorial group theory;
probabilities on groups;
computational aspects and complexity;
harmonic and functional analysis on groups, free probability;
ergodic theory of group actions;
cohomology of groups and exotic cohomologies;
groups and low-dimensional topology;
group actions on trees, buildings, rooted trees.
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