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This article is cited in 1 scientific paper (total in 1 paper)
Quasirecognition by Prime Graph of $^2D_{n}(3^\alpha)$ where $n=4m+1\ge 21$ and $\alpha$ is Odd
A. Babai, B. Khosravi Amirkabir University of Technology, Iran
Abstract:
Let $G$ be a finite group. The prime graph of $G$ is denoted by $\Gamma(G)$. In this paper, as the main result, we show that if $G$ is a finite group such that $\Gamma(G)=\Gamma(^2D_n(3^\alpha))$, where $n=4m+1$ and $\alpha$ is odd, then $G$ has a unique non-Abelian composition factor isomorphic to $^2D_n(3^\alpha)$. We also show that if $G$ is a finite group satisfying $|G|=|^2D_n(3^\alpha)|$, and $\Gamma(G)=\Gamma(^2D_n(3^\alpha))$, then $G\cong{}^2D_n(3^\alpha)$. As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for $^2D_n(3^\alpha)$. Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that $^2D_n(3^\alpha)$ is quasirecognizable by the spectrum.
Keywords:
prime graph, simple group, recognition, quasirecognition.
Received: 28.07.2012
Citation:
A. Babai, B. Khosravi, “Quasirecognition by Prime Graph of $^2D_{n}(3^\alpha)$ where $n=4m+1\ge 21$ and $\alpha$ is Odd”, Mat. Zametki, 95:3 (2014), 323–334; Math. Notes, 95:3 (2014), 293–303
Linking options:
https://www.mathnet.ru/eng/mzm10421https://doi.org/10.4213/mzm10421 https://www.mathnet.ru/eng/mzm/v95/i3/p323
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Abstract page: | 496 | Full-text PDF : | 146 | References: | 73 | First page: | 24 |
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