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Non-commuting graph of a group. (English) Zbl 1105.20016

The non-commuting graph \(\Gamma_G\) of a non-Abelian group \(G\) is defined as follows. The vertex set of \(\Gamma_G\) is \(V(G)=G-Z(G)\) and two vertices \(x\) and \(y\) are joined by an edge if and only if \(xy\neq yx\). This graph was first defined by P. Erdős which is quoted by B. H. Neumann [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. A natural question to ask is how the graph theoretical properties of \(\Gamma_G\) are related to the group theoretical properties of \(G\).
In the paper under review the authors answer some questions about \(\Gamma_G\) and relate them to the structure of \(G\). But the bulk of the paper is centered around the verification of the following Conjecture: Let \(G\) and \(H\) be two non-Abelian groups with the property that \(\Gamma_G \) and \(\Gamma_H\) are isomorphic graphs, then \(|G|=|H|\), and if \(G\) is a simple group \(G\cong H\).
The authors prove the first part of the conjecture for the groups \(G\cong S_n\), \(A_n\), \(\text{PSL}(2,q)\), \(D_n\) or a non-solvable AC-group, and the second part for the groups \(G\cong\text{PSL}(2,2^n)\) and the Suzuki groups \(^2B_2(2^{2n+1})\), \(n>1\). Some invariants of the graph \(\Gamma_G\), such as the clique number, chromatic number, etc., are found for special groups \(G\).

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20D06 Simple groups: alternating groups and groups of Lie type

Citations:

Zbl 0333.05110
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References:

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