Virtual endomorphisms of the group $pg
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...
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Published in | Researches in mathematics (Online) Vol. 32; no. 1; pp. 3 - 15 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Oles Honchar Dnipro National University
08.07.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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 $2$. |
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ISSN: | 2664-4991 2664-5009 |
DOI: | 10.15421/242401 |