TY - JOUR

T1 - On a commuting graph on conjugacy classes of groups

AU - Herzog, Marcel

AU - Longobardi, Patrizia

AU - Maj, Mercede

N1 - Funding Information:
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.

PY - 2009/10

Y1 - 2009/10

N2 - 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.

AB - 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.

KW - Conjugacy classes

KW - Graphs

KW - Locally finite groups

KW - Periodic groups

UR - http://www.scopus.com/inward/record.url?scp=70449486710&partnerID=8YFLogxK

U2 - 10.1080/00927870802502779

DO - 10.1080/00927870802502779

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AN - SCOPUS:70449486710

VL - 37

SP - 3369

EP - 3387

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 10

ER -