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Noncommuting graph characterization of some simple groups with connected prime graphs. (English) Zbl 1176.20013

The noncommuting graph \(\nabla(G)\) associated with a nonabelian finite group \(G\) is defined as follows: the vertex set of \(\nabla(G)\) is \(G\setminus Z(G)\), and two vertices are adjacent by an edge whenever they do not commute. A. Abdollahi, S. Akbari and H. R. Maimani [J. Algebra 298, No. 2, 468-492 (2006; Zbl 1105.20016)] conjectured that if \(M\) is a finite nonabelian simple group and \(G\) is a group such that \(\nabla(G)\cong\nabla(M)\), then \(G\cong M\). Even though this conjecture is known to hold for all simple groups with nonconnected prime graphs and the alternating group \(A_{10}\) (see the paper by L.-L. Wang and W.-J. Shi [Commun. Algebra 36, No. 2, 523-528 (2008; Zbl 1153.20013)]), it is still unknown for all simple groups. In the present paper, the authors prove that the conjecture is also true for the group \(L_4(8)\). They remark that the new method used in this paper also works well in the case of \(L_4(4)\), \(L_4(7)\), \(U_4(7)\), etc.

MSC:

20D05 Finite simple groups and their classification
20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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