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Quasirecognition by Prime Graph of ²D_p(3) where p = 2ⁿ + 1 ≥ 5 is a Prime

Babai, A., and Khosravi, B., and Hasani, N., (2009) Quasirecognition by Prime Graph of ²D_p(3) where p = 2ⁿ + 1 ≥ 5 is a Prime. Bulletin of the Malaysian Mathematical Sciences Society, 32 (3). pp. 343-350. ISSN 0126-6705

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Official URL: http://math.usm.my/bulletin/pdf/v32n3/v32n3p8.pdf

Affiliations

Amirkabir University of Technology (Tehran Polytechnic), Iran. Faculty of Mathematics and Computer Science. Dept. of Pure Mathematics.
Amirkabir University of Technology (Tehran Polytechnic), Iran. Faculty of Mathematics and Computer Science. Dept. of Pure Mathematics & Institute for Research in Fundamental Sciences (IPM), Iran. School of Mathematics.
Amirkabir University of Technology (Tehran Polytechnic), Iran. Faculty of Mathematics and Computer Science. Dept. of Pure Mathematics.

Abstract

In this paper as the main result, we show that if $G$ is a finite group such that $\Gamma (G) = \Gamma (^2 D_{p}(3))$, where $p = 2^{n} + 1$, $(n \geq 2)$ is a prime number, then $G$ has a unique non-abelian composition factor isomorphic to $(^2 D_{p}(3))$. We also show that if $G$ is a finite group satisfying $\vert G \vert = \vert^2D_{p}(3) \vert$ and $\Gamma (G) = \Gamma (^2D_{p}(3))$, then $G \cong ^2 D_{p}(3)$. As a consequence of our result we give a new proof for a conjecture of W. J. Shi and J. X. Bi [A characteristic property for each finite projective special linear group, in Groups—Canberra 1989, 171–180, Lecture Notes in Math., 1456, Springer, Berlin] for $^2 D_{p}(3)$. Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered.

2000 Mathematics Subject Classification: 20D05, 20D60

Item Type:Journal
Additional Information:The authors express their gratitude to the referee. The second author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, IRAN for the financial support. The second author was supported in part by a grant from IPM (No. 87200022).
Keywords:Quasirecognition, prime graph, simple group, element order.
Subjects:Q Science, Computer Science
ID Code:7815

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