On a Commuting Graph on Conjugacy Classes of Groups

Abstract

Abstract We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8. Key Words: Conjugacy classesGraphsLocally finite groupsPeriodic groups2000 Mathematics Subject Classification: Primary 20D60, 20E45Secondary 20F16 ACKNOWLEDGMENTS The authors are grateful to Zvi Arad for his constructive remarks concerning the graph Π(G). They are also grateful to the referee for his important comments and in particular, for his contribution to the proof of Theorem 19. The first author is grateful to the Department of Mathematics and Informatics of the University of Salerno for its hospitality and support, while this investigation was carried out. Notes Communicated by M. Dixon. Dedicated to the memory of our dear colleague and friend Silvia Lucida.