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Extending results of Morgan and Parker about commuting graphs. (English) Zbl 1517.20034

Let \(G\) be a finite group; the commuting graph of \(G\) is the graph \(\Gamma (G)\) whose vertex set is \(G\setminus Z(G)\), that is the set of all non central elements of \(G\), and two vertices \(x\) and \(y\) are adjacent if and only if \([x,y]=1\), that is if and only if they commute.
Commuting graphs were studied by many authors; much of the research regarding this subject is related to simple groups. For instance, R. Solomon and A. Woldar [J. Group Theory 16, 793–824 (2013; Zbl 1293.20024)] proved that simple groups are characterized by their commuting graph.
A. Iranmanesh and A. Jafarzadeh [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 23, No. 1, 7–13 (2007; Zbl 1135.20014)] conjectured that there is a universal bound on the diameter of commuting graphs but M. Giudici and C. Parker [J. Comb. Theory, Ser. A 120, No. 7, 1600–1603 (2013; Zbl 1314.05055)] constructed a family of 2-groups, nilpotent of class 2, for which there is no bound on the diameter of the commuting graphs.
On the other hand, C. Parker [Bull. Lond. Math. Soc. 45, No. 4, 839–848 (2013; Zbl 1278.20017)] proved that the commuting graph of a solvable group \(G\) with trivial center is disconnected if and only if \(G\) is a Frobenius group or it has normal subgroups \(K\le L\) such that \(L\) and \(G/K\) are Frobenius groups with Frobenius kernels \(K\) and \(L/K\), respectively (that is it is a 2-Frobenius group). Moreover, when \(\Gamma (G)\) is connected it has diameter at most 8.
Afterwards G. L. Morgan and C. W. Parker [J. Algebra 393, 41–59 (2013; Zbl 1294.20033)] removed the solvability hypothesis on \(G\) and proved that if \(G\) is any group with trivial center, then all the connected components of \(\Gamma(G)\) have diameter at most 10.
In this paper, the authors try to extend results of Parker [loc. cit,] and Parker and Morgan [loc. cit,] and they show that it is possible to replace the hypothesis that \(Z(G)=1\) with the hypothesis that \(G'\cap Z(G)=1\). The main result is the following
Theorem 1.1 Let \(G\) be a group and suppose that \(G'\cap Z(G)=1\); then
1) \(\Gamma(G)\) is connected if and only if \(\Gamma (G/Z(G))\) is connected;
2) every connected component of \(\Gamma(G)\) has diameter at most \(10\);
3) if \(G\) is solvable and \(\Gamma (G)\) is connected, then \(\Gamma (G)\) has diameter at most \(8\);
4) if \(G\) is solvable, then \(\Gamma (G)\) is disconnected if and only if \(G/Z(G)\) is either a Frobenius group or a \(2\)-Frobenius group.
The hypothesis that \(G'\cap Z(G)=1\) can be relaxed, in fact, it suffices to assume that, for all \(x,y\in G\), the commutator \([x,y]\in Z(G)\) if and only if \([x,y]=1\).
A class of finite groups satisfying the hypothesis of Theorem 1.1 is, for instance, the class of \(A\)-groups. In fact, if every Sylow subgroup of a group \(G\) is abelian (that is \(G\) is an \(A\)-group), then it is possible to prove that \(G'\cap Z(G)=1\)
Finally, the authors present some examples to clarify various points.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20E34 General structure theorems for groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)

Software:

Magma; GAP
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References:

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