Geodesic
Quasirecognition by Prime Graph of $^2D_{n}(3^\alpha)$ where $n=4m+1\ge 21$ and~$\alpha$ is Odd
Matematičeskie zametki, Tome 95 (2014) no. 3, pp. 323-334

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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, recognition, quasirecognition.
Mots-clés : simple group 
@article{MZM_2014_95_3_a0,
     author = {A. Babai and B. Khosravi},
     title = {Quasirecognition by {Prime} {Graph} of $^2D_{n}(3^\alpha)$ where $n=4m+1\ge 21$ and~$\alpha$ is {Odd}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {323--334},
     publisher = {mathdoc},
     volume = {95},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_3_a0/}
}
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A. Babai; B. Khosravi. Quasirecognition by Prime Graph of $^2D_{n}(3^\alpha)$ where $n=4m+1\ge 21$ and~$\alpha$ is Odd. Matematičeskie zametki, Tome 95 (2014) no. 3, pp. 323-334. http://geodesic.mathdoc.fr/item/MZM_2014_95_3_a0/