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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 $xy\ne yx$. This graph was first defined by P. Erdős which is quoted by {\it 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 $G$.

20D60Arithmetic and combinatorial problems on finite groups
05C25Graphs and abstract algebra
20D06Simple groups: alternating groups and groups of Lie type
Full Text: DOI
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