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On a Graph Related to Conjugacy Classes of Groups
Author(s) -
Bertram Edward A.,
Herzog Marcel,
Mann Avinoam
Publication year - 1990
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/22.6.569
Subject(s) - mathematics , conjugacy class , combinatorics , coprime integers , finite group , abelian group , graph , simple graph , simple group , discrete mathematics , simple (philosophy) , group (periodic table) , philosophy , chemistry , organic chemistry , epistemology
Let G be a finite group. Attach to G the following graph Γ: its vertices are the non‐central conjugacy classes of G , and two vertices are connected if their cardinalities are not coprime. Denote by n (Γ) the number of the connecte components of Γ. We prove that n (Γ) ⩽ 2 for all finite groups, and we completely characterize groups with n (Γ) = 2. When Γ is connected, then the diameter of the graph is at most 4. For simple non‐abelian finite groups, the graph is complete. Similar results are proved for infinite FC‐groups.