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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.
UDC:
512.542
MSC: 05C25,
20D05,
20D06 Received: 10.02.2016
DOI:
10.21538/0134-4889-2016-22-3-101-116