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This article is cited in 2 scientific papers (total in 2 papers)
On finite simple classical groups over fields of different characteristics with coinciding prime graphs
M. R. Zinov'evaab a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
Suppose that $G$ is a finite group, $\pi(G)$ is the set of prime divisors of its order, and $\omega(G)$ is the set of orders of its elements. We define a graph on $\pi(G)$ with the following adjacency relation: different vertices $r$ and $s$ from $\pi(G)$ are adjacent if and only if $rs\in \omega(G)$. This graph is called the $\it{Gruenberg-Kegel\, graph }$ for the $\it{prime\, graph }$ of $G$ and is denoted by $GK(G)$. Let $G$ and $G_1$ be two nonisomorphic finite simple groups of Lie type over fields of orders $q$ and $q_1$, respectively, with different characteristics. It is proved that, if $G$ is a classical group of a sufficiently high Lie rank, then the prime graphs of the groups $G$ and $G_1$ may coincide only in one of three cases. It is also proved that, if $G=A_1(q)$ and $G_1$ is a classical group, then the prime graphs of the groups $G$ and $G_1$ coincide only if $\{G,G_1\}$ is equal to $\{A_1(9),A_1(4)\}$, $\{A_1(9),A_1(5)\}$, $\{A_1(7),A_1(8)\}$, or $\{A_1(49),^2A_3(3)\}$.
Keywords:
finite simple classical group, prime graph, spectrum.
Received: 10.02.2016
Citation:
M. R. Zinov'eva, “On finite simple classical groups over fields of different characteristics with coinciding prime graphs”, Trudy Inst. Mat. i Mekh. UrO RAN, 22, no. 3, 2016, 101–116; Proc. Steklov Inst. Math. (Suppl.), 297, suppl. 1 (2017), 223–239
Linking options:
https://www.mathnet.ru/eng/timm1325 https://www.mathnet.ru/eng/timm/v22/i3/p101
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Abstract page: | 255 | Full-text PDF : | 68 | References: | 48 | First page: | 2 |
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