M. R. Zinov'eva. On finite simple linear and unitary groups of small size over fields of different characteristics with coinciding prime graphs

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. A graph with the following adjacency relation is defined on $\pi(G)$: different vertices $r$ and $s$ from $\pi(G)$ are adjacent if and only if $rs\in \omega(G)$. This graph is called the Gruenberg-Kegel graph or the prime graph of $G$ and is denoted by $GK(G)$. In A. V. Vasil'ev's Question 16.26 from the "Kourovka Notebook," it is required to describe all pairs of nonisomorphic simple nonabelian groups with identical Gruenberg-Kegel graphs. M. Hagie and M. A. Zvezdina gave such a description in the case where one of the groups coincides with a sporadic group and an alternating group, respectively. The author solved this question for finite simple groups of Lie type over fields of the same characteristic. In the present paper we prove the following theorem.

Theorem.  Let $G=A_{n-1}^{\pm}(q)$, where $n\in\{3,4,5,6\}$, and let $G_1$ be a finite simple group of Lie type over a field of order $q_1$ nonisomorphic to $G$, where $q=p^f$, $q_1=p_1^{f_1}$, and $p$ and~$p_1$ are different primes. If the graphs $GK(G)$ and $GK(G_1)$ coincide, then one of the following statements holds:

$(1)$ $\{G,G_1\}=\{A_1(7),A_1(8)\}$;

$(2)$ $\{G,G_1\}=\{A_3(3),{^2}F_4(2)'\}$;

$(3)$ $\{G,G_1\}=\{{^2}A_3(3),A_1(49)\}$;

$(4)$ $\{G,G_1\}=\{A_2(q),{^3}D_4(q_1)\}$, where $(q-1)_3\neq 3$, $q+1\neq 2^k$, and $q_1>2$;

$(5)$ $\{G,G_1\}=\{A_4^{\varepsilon}(q),A_4^{\varepsilon_1}(q_1)\}$, where $qq_1$ is odd;

$(6)$ $\{G,G_1\}=\{A_4^{\varepsilon}(q),{^3}D_4(q_1)\}$, where $(q-\epsilon1)_5\neq 5$ and $q_1>2$;

$(7)$ $G=A_5^{\varepsilon}(q)$ and $G_1\in\{B_3(q_1),C_3(q_1),D_4(q_1)\}$.

Keywords: finite simple group of Lie type, prime graph, Gruenberg--Kegel graph, spectrum.

The paper was received by the Editorial Office on July 10, 2018.

Marianna Rifkhatovna Zinov’eva, Cand. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Ekaterinburg, 620002 Russia, e-mail: zinovieva-mr@yandex.ru

Funding Agency: This work was supported by the Integrated Program for Fundamental Research of the Ural Branch of the Russian Academy of Sciences (project no. 18-1-1-17) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).


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