The virtually generating graph of a profinite group. Issue 4 (15th December 2021)

• Record Type: Journal Article
• Authors: Lucchini, Andrea
• Is Part Of: Canadian mathematical bulletin =. Volume 64:Issue 4(2021)
• Page Start: 808
• Page End: 819
• Publication Date: 2021-12-15
• Subjects: 20D60 -- 20F05 -- 05C25; Group generation -- generating graph -- profinite groups; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/S0008439520000843
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: We consider the graph $\Gamma _{\text {virt}}(G)$ 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 $\Delta _{\text {virt}}(G)$ obtained from $\Gamma _{\text {virt}}(G)$ 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 $\Delta _{\operatorname {\mathrm {virt}}}(G)$ has precisely t connected components. Moreover, we study the graph $\widetilde \Gamma _{\operatorname {\mathrm {virt}}}(G)$, 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 $\widetilde \Delta _{\operatorname {\mathrm {virt}}}(G)$ obtained removing the isolated vertices is connected and has diameter at most 3.