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Quasirecognition by prime graph of \(^2D_p(3)\) where \(p=2^n+1\geq 5\) is a prime. (English) Zbl 1172.20015
Summary: As the main result, we show that if \(G\) is a finite group such that \(\Gamma(G) =\Gamma(^2D_p(3))\), where \(p=2^n+1\), (\(n\geq 2\)) is a prime number, then \(G\) has a unique non-Abelian composition factor isomorphic to \(^2D_p(3)\). We also show that if \(G\) is a finite group satisfying \(|G|=|^2D_p(3)|\) and \(\Gamma(G)=\Gamma(^2D_p(3))\), then \(G\cong{^2D_p(3)}\). As a consequence of our result we give a new proof for a conjecture of W.-J. Shi and J.-X. Bi [Lect. Notes Math. 1456, 171-180 (1990; Zbl 0718.20009)] for \(^2D_p(3)\). Applications of this result to the problem of recognition of finite simple groups by the set of element orders are also considered.

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.)
Zbl 0718.20009
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