2008/10/02 by Aliréza Abdollahi, Alireza Abdollahi, Abdollahi, Alireza +2
Mathematics · #05C25 #20D60 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras #math.CO #math.GR #msc:05C25 #msc:20D60
paper · pdf · doi:10.48550/arxiv.0810.0345
to appear in Journal of Algebra and its Applications
arxiv created 2008/10/02 · openalex publication_date 2008/10/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We associate a graph CG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\backslash Cyc(G) as vertex set, where Cyc(G)=\x∈ G | < x,y> is cyclic for all y∈ G\ is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever w(CG)<31 and the equality for a non-solvable group G holds if and only if G/Cyc(G)≅ A5 or S5.