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 {\it 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 finite simple nonabelian groups with identical Gruenberg-Kegel graphs. M. Hagie (2003) and M.A. Zvezdina (2013) gave such a description in the case where one of the groups coincides with a sporadic group and an alternating group, respectively. The author (2014) solved this question for pairs finite simple groups of Lie type over fields of the same characteristic. In the present paper we prove the following theorem.
Theorem. Let $G$ be a finite simple group of exceptional Lie type over a field with $q$ elements, and let $G_1$ be a finite simple group of Lie type over a field with $q$ elements nonisomorphic to $G$, where $q$ and $q_1$ are coprime. If $GK(G)=GK(G_1)$, then one of the following statements holds:
$(1)$ $\{G,G_1\}=\{G_2(3),A_1(13)\}$;
$(2)$ $\{G,G_1\}=\{{^2}F_4(2)',A_3(3)\}$;
$(3)$ $\{G,G_1\}=\{{^3}D_4(q),A_2(q_1)\}$, where $(q_1-1)_3\neq 3$ and $q_1+1\neq 2^{k_1}$;
$(4)$ $\{G,G_1\}=\{{^3}D_4(q),A_4^{\pm}(q_1)\}$, where $(q_1\mp1)_5\neq 5$;
$(5)$ $\{G,G_1\}=\{G_2(q),G_2(q_1)\}$, where $q$ and $q_1$ are not powers of the number~3;
$(6)$ $\{G,G_1\}$ is one of the pairs $\{F_4(q),F_4(q_1)\}$, $\{{^3}D_4(q),{^3}D_4(q_1)\}$, and $\{E_8(q),E_8(q_1)\}$.
The existence of pairs of groups in statements (3)-(6) is unknown.
Keywords: finite simple exceptional group of Lie type, spectrum, prime graph
Received April 3, 2020
Revised May 11, 2020
Accepted May 25, 2020
Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00456) 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).
Marianna Rifkhatovna Zinov’eva, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Ekaterinburg, 620083 Russia, e-mail: zinovieva-mr@yandex.ru
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Cite this article as: M.R. Zinov’eva. On finite simple groups of exceptional Lie type over fields of different characteristics with coinciding prime graphs, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 2, pp. 147–160; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 313, Suppl. 1, pp. S228–S240.
