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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 $V\left(G\right)=G-Z\left(G\right)$ and two vertices $x$ and $y$ are joined by an edge if and only if $xy\ne yx$. 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 $G\cong {S}_{n}$, ${A}_{n}$, $\text{PSL}\left(2,q\right)$, ${D}_{n}$ or a non-solvable AC-group, and the second part for the groups $G\cong \text{PSL}\left(2,{2}^{n}\right)$ and the Suzuki groups ${}^{2}{B}_{2}\left({2}^{2n+1}\right)$, $n>1$. 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 on finite groups 05C25 Graphs and abstract algebra 20D06 Simple groups: alternating groups and groups of Lie type
##### Keywords:
non-commuting graphs; finite simple groups