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Quasirecognition by prime graph of finite simple groups \(L_n(2)\) and \(U_n(2)\). (English) Zbl 1232.20020

Let \(G\) be a finite group and the prime graph \(\Gamma(G)\) of \(G\) is defined as follows: the vertices of \(\Gamma(G)\) are the primes dividing the order \(|G|\) and two distinct vertices \(p,q\) are joined by an edge if \(G\) has an element of order \(pq\) [see J. S. Williams, J. Algebra 69, 487-513 (1981; Zbl 0471.20013)]. A finite non-Abelian simple \(S\) is said to be quasirecognizable by prime graph if \(G\) with \(\Gamma(G)=\Gamma(S)\) has a unique nonabelian composition factor isomorphic to \(S\). In this paper, the authors prove that the simple groups \(L_n(2)\) and \(U_n(2)\) are quasirecognizable. As a consequence of the above result the authors give a new proof for the recognition by element orders of \(L_n(2)\).

MSC:

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

Citations:

Zbl 0471.20013
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References:

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