2018/10/08 by Cristina Acciarri, Acciarri, Cristina, Andrea Lucchini +1
Mathematics · #05C25 #20D60 #20K05 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1810.03508
openalex publication_date 2018/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a group G, let Γ(G) denote the graph defined on the elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Moreover let Γ^*(G) be the subgraph of Γ(G) that is induced by all the vertices of Γ(G) that are not isolated. We prove that if G is a 2-generated non-cyclic abelian group then Γ^*(G) is connected. Moreover diam(Γ^*(G))=2 if the torsion subgroup of G is non-trivial and diam(Γ^*(G))=∞ otherwise. If F is the free group of rank 2, then Γ^*(F) is connected and we deduce from diam(Γ^*(ℤ× ℤ))=∞ that diam(Γ^*(F))=∞.