| Autor: |
Lucchini, Andrea |
| Rok vydání: |
2020 |
| Předmět: |
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| Zdroj: |
Can. Math. Bull. 64 (2021) 808-819 |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.4153/S0008439520000843 |
| Popis: |
We consider the graph $\Gamma_{\rm{virt}}(G)$ whose vertices are the elements of a finitely generated profinite group $G$ and where two vertices $x$ and $y$ are adjacent if and only if they topologically generate an open subgroup of $G$. We investigate the connectivity of the graph $\Delta_{\rm{virt}}(G)$ obtained from $\Gamma_{\rm{virt}}(G)$ by removing its isolated vertices. In particular we prove that for every positive integer $t$, there exists a finitely generated prosoluble group $G$ with the property that $\Delta_{\rm{virt}}(G)$ has precisely $t$ connected components. Moreover we study the graph $\tilde \Gamma_{\rm{virt}}(G)$, whose vertices are again the elements of $G$ and where two vertices are adjacent if and only if there exists a minimal generating set of $G$ containing them. In this case we prove that the subgraph $\tilde \Delta_{\rm{virt}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3. |
| Databáze: |
arXiv |
| Externí odkaz: |
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