2025/11/20 by Alland, Lucas, Fridman, Andrei, Keller, Thomas Michael
Mathematics · #05C25 #20D60 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2511.16403
openalex publication_date 2025/11/20 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28
Given a finite group G, its prime graph Γ(G) (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of |G| and where edges \p, q\ exist whenever G contains an element of order pq. We continue the study of prime graphs for T-solvable groups; that is, groups whose composition factors are either abelian or isomorphic to some fixed non-abelian simple group T. For a large class of non-abelian simple groups T, we prove that the prime graph complements of T-solvable groups are always realizable by a solvable group and a quasi simple or almost simple T-solvable group acting by automorphisms on a direct product of elementary abelian groups. We conjecture that a similar result holds in full generality. Moreover, we apply our result to classify in purely graph-theoretic terms the prime graph complements of PSL(2,13)-solvable groups, and indicate other interesting classes of groups matching the assumptions of our main theorem.