{"title":"群$pg$的虚同构","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":"{\"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}","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}