{"title":"Virtual endomorphisms of the group $pg$","authors":"I. Bondarenko, D. Zashkolny","doi":"10.15421/242401","DOIUrl":null,"url":null,"abstract":"A virtual endomorphism of a group $G$ is a homomorphism of the form $\\phi:H\\rightarrow G$, where $H<G$ is a subgroup of finite index. A virtual endomorphism $\\phi:H\\rightarrow G$ is called simple if there are no nontrivial normal $\\phi$-invariant subgroups, that is, the $\\phi$-core is trivial. We describe all virtual endomorphisms of the plane group $pg$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $pg$ admits a transitive self-similar (as well as finite-state) action of degree $d$ if and only if $d\\geq 2$ is not an odd prime, and admits a self-replicating action of degree $d$ if and only if $d\\geq 6$ is not a prime or a power of $2$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":" 719","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Researches in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.15421/242401","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
A virtual endomorphism of a group $G$ is a homomorphism of the form $\phi:H\rightarrow G$, where $H
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