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 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 of 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 holds: \((1)\ \{G,G_{1}\}=\{G_{2}(3),A_{1}(13)\}\); \((2)\ \{G,G_{1}\}=\{{{}^{2}}F_{4}(2)^{\prime},A_{3}(3)\}\); \((3)\ \{G,G_{1}\}=\{{{}^{3}}D_{4}(q),A_{2}(q_{1})\}\), 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 \((q_{1}\mp 1)_{5}\neq 5\); \((5)\ \{G,G_{1}\}=\{G_{2}(q),G_{2}(q_{1})\}\), where \(q\) and \(q_{1}\) are not powers of 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.

假设 \(G\)是一个有限群，\(\pi(G)\)是其阶的素因数集，而\(\omega(G)\)是其元素的阶集。用下面的邻接关系的曲线图上定义 \（\ PI（G）\） ：不同顶点 \（R \）和 \（S \）从 \（\ PI（G）\）相邻当且仅当 \ (rs\in\omega(G)\)。该图称为Gruenberg-Kegel 图或\(G\)的素数图， 用 \(GK(G)\) 表示。在 AV Vasil'ev 的问题 16.26 中来自Kourovka Notebook，需要用相同的 Gruenberg-Kegel 图来描述所有非同构有限简单非阿贝尔群对。M. Hagie (2003) 和 MA Zvezdina (2013) 在其中一组分别与零星组和交替组重合的情况下给出了这样的描述。作者 (2014) 在相同特征的域上解决了李型有限单群对的问题。在本文中，我们证明以下定理。定理。令 \(G\)是具有\(q\) 个元素的域上的异常Lie 型有限单群 ，令 \(G_{1}\)是具有 \( q\)个与\(G\)非同构的元素 ，其中 \(q\)和 \(q_{1}\)是互质的。如果\(GK(G)=GK(G_{1})\)，则以下之一成立： \((1)\ \{G,G_{1}\}=\{G_{2}(3 ),A_{1}(13)\}\) ; \((2)\ \{G,G_{1}\}=\{{{}^{2}}F_{4}(2)^{\prime},A_{3}(3)\}\ ) ; \((3)\ \{G,G_{1}\}=\{{{}^{3}}D_{4}(q),A_{2}(q_{1})\}\) ,其中\((q_{1}-1)_{3}\neq 3\)和\(q_{1}+1\neq 2^{k_{1}}\) ; \((4)\ \{G,G_{1}\}=\{{{}^{3}}D_{4}(q),A_{4}^{\pm}(q_{1}) \}\) , 其中\((q_{1}\mp 1)_{5}\neq 5\) ; \((5)\ \{G,G_{1}\}=\{G_{2}(q),G_{2}(q_{1})\}\)，其中 \(q\)和 \ (q_{1}\)不是 3 的幂；\((6)\ \{G,G_{1}\}\)是\(\{F_{4}(q),F_{4}(q_{1})\}\) 对之一，\(\{{{}^{3}}D_{4}(q),{{}^{3}}D_{4}(q_{1})\}\)和\(\{E_{ 8}(q),E_{8}(q_{1})\}\) . 语句 (3)-(6) 中是否存在成对的群是未知的。