On a commuting graph on conjugacy classes of groups

Marcel Herzog, Patrizia Longobardi, Mercede Maj

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)3369-3387
Number of pages19
JournalCommunications in Algebra
Volume37
Issue number10
DOIs
StatePublished - Oct 2009

Keywords

  • Conjugacy classes
  • Graphs
  • Locally finite groups
  • Periodic groups

Fingerprint

Dive into the research topics of 'On a commuting graph on conjugacy classes of groups'. Together they form a unique fingerprint.

Cite this