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On the Diameter of a Graph Related to Conjugacy Classes of Groups
Author(s) -
Chillag David,
Herzog Marcel,
Mann Avinoam
Publication year - 1993
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/25.3.255
Subject(s) - mathematics , combinatorics , conjugacy class , graph , finite group , connected component , discrete mathematics , group (periodic table) , chemistry , organic chemistry
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 co‐prime. Denote by n (Γ) the number of the connected components of Γ. By [1], n (Γ) ⩽ 2 for all finite groups, and if Γ is connected, the diameter of the graph is at most 4. In this paper we prove that if Γ is connected, then the diameter of the graph is at most 3, and this bound is the best possible. Similar results are proved for infinite FC‐groups.