The diameter of the commuting graph of a finite group with trivial centre. (English) Zbl 1294.20033
The commuting graph \(\Gamma(G)\) of a group \(G\) is the graph which has vertices the non-central elements of \(G\) and two distinct vertices of \(\Gamma(G)\) are adjacent if and only if they commute in \(G\).
A. Iranmanesh and A. Jafarzadeh [J. Algebra Appl. 7, No. 1, 129-146 (2008; Zbl 1151.20013)] demonstrated that the commuting graphs of the finite symmetric and alternating groups are either disconnected or have diameter at most 5 and conjectured that there is an absolute upper bound for the diameter of a connected commuting graph of a non-Abelian finite group. M. Giudichi and C. Parker [J. Comb. Theory, Ser. A 120, No. 7, 1600-1603 (2013)] have shown that this conjecture is incorrect by constructing an infinite family of special 2-groups with commuting graphs of increasing diameter. In this paper, the authors prove that the connected components of the commuting graph of a finite group with trivial centre have diameter at most 10.
Various special kinds of finite groups with bounded diameter commuting graph were investigated by Y. Segev and G. M. Seitz [Pac. J. Math. 202, No. 1, 125-225 (2002; Zbl 1058.20015)], Iranmanesh and Jafarzadeh [loc. cit.], T. J. Woodkock [Commuting graphs of finite groups. PhD thesis, Univ. Virginia (2010)], M. Giudici and A. Pope [Australas. J. Comb. 48, 221-230 (2010; Zbl 1232.05114); J. Group Theory 17, No. 1, 131-149 (2014)], the second author [Bull. Lond. Math. Soc. 45, No. 4, 839-848 (2013; Zbl 1278.20017)].
A. Iranmanesh and A. Jafarzadeh [J. Algebra Appl. 7, No. 1, 129-146 (2008; Zbl 1151.20013)] demonstrated that the commuting graphs of the finite symmetric and alternating groups are either disconnected or have diameter at most 5 and conjectured that there is an absolute upper bound for the diameter of a connected commuting graph of a non-Abelian finite group. M. Giudichi and C. Parker [J. Comb. Theory, Ser. A 120, No. 7, 1600-1603 (2013)] have shown that this conjecture is incorrect by constructing an infinite family of special 2-groups with commuting graphs of increasing diameter. In this paper, the authors prove that the connected components of the commuting graph of a finite group with trivial centre have diameter at most 10.
Various special kinds of finite groups with bounded diameter commuting graph were investigated by Y. Segev and G. M. Seitz [Pac. J. Math. 202, No. 1, 125-225 (2002; Zbl 1058.20015)], Iranmanesh and Jafarzadeh [loc. cit.], T. J. Woodkock [Commuting graphs of finite groups. PhD thesis, Univ. Virginia (2010)], M. Giudici and A. Pope [Australas. J. Comb. 48, 221-230 (2010; Zbl 1232.05114); J. Group Theory 17, No. 1, 131-149 (2014)], the second author [Bull. Lond. Math. Soc. 45, No. 4, 839-848 (2013; Zbl 1278.20017)].
Reviewer: Anatoli Kondrat’ev (Ekaterinburg)
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D05 | Finite simple groups and their classification |
| 05C40 | Connectivity |
Software:
MagmaReferences:
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