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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.

##### MSC:
 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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