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A characterization of the groups \(\text{PSU}(4,4)\) and \(\text{PSL}(4,4)\) by non-commuting graph. (English) Zbl 1219.20013

Let \(G\) be a finite group. The graph \(\Gamma_G\) has the vertex set \(G-Z(G)\) and two vertices \(a,b\) are adjacent whenewer \(ab\neq ba\).
Conjecture 2. Let \(S\) be a finite nonabelian simple group. If \(G\) is a group such that \(\Gamma_G\simeq\Gamma_S\) then \(G\simeq S\).
In this paper it is proved that Conjecture 2 is valid for \(L_4(4)\) and for \(U_4(4)\).

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20G40 Linear algebraic groups over finite fields
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