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Trudy Inst. Mat. i Mekh. UrO RAN, 2016, Volume 22, Number 3, Pages 101–116 (Mi timm1325)  

This article is cited in 1 scientific paper (total in 1 paper)

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.

Funding Agency Grant Number
Russian Science Foundation 15-11-10025


DOI: https://doi.org/10.21538/0134-4889-2016-22-3-101-116

Full text: PDF file (268 kB)
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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2017, 297, suppl. 1, 223–239

Bibliographic databases:

Document Type: Article
UDC: 512.542
MSC: 05C25, 20D05, 20D06
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

Citation in format AMSBIB
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\paper On finite simple classical groups over fields of different characteristics with coinciding prime graphs
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2016
\vol 22
\issue 3
\pages 101--116
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\crossref{https://doi.org/10.21538/0134-4889-2016-22-3-101-116}
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\jour Proc. Steklov Inst. Math. (Suppl.)
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\vol 297
\issue , suppl. 1
\pages 223--239
\crossref{https://doi.org/10.1134/S0081543817050248}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. R. Zinoveva, “O konechnykh prostykh lineinykh i unitarnykh gruppakh nad polyami raznykh kharakteristik, grafy prostykh chisel kotorykh sovpadayut. I”, Tr. IMM UrO RAN, 23, no. 4, 2017, 136–151  mathnet  crossref  elib
  • Trudy Instituta Matematiki i Mekhaniki UrO RAN
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