Suppose that G is a finite group, π(G) is the set of prime divisors of its order, and ω(G) is the set of orders of its elements. A graph with the following adjacency relation is defined on π(G): different vertices r and s from π(G) are adjacent if and only if rs ∈ ω(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\(G = A_{n - 1}^ \pm \left( q \right)\), where n ∈{3, 4, 5, 6}, and let G₁be a finite simple group of Lie type over a field of order q₁nonisomorphic to G, where q = p^f, \({q_1} = p_1^{{f_1}}\), and p and p₁are different primes. If the graphs GK(G) and GK(G₁) coincide, then one of the following statements holds(1) {G, G₁} = {A₁(7), A₁(8)}(2) {G, G₁} = {A₃(3), ²F₄(2)′}(3) {G, G₁} = {²A₃(3), A₁(49)}(4) {G, G₁} = {A₂(q), ³D₄(q₁)}, where (q − 1)₃ ≠ 3, q + 1 ≠ 2^k, and q₁ > 2(5) \(\left\{G, G_{1}\right\}=\left\{A_{4}^{\varepsilon}(q), A_{4}^{\varepsilon_{1}}\left(q_{1}\right)\right\}\), where qq₁is odd(6) \(\left\{ {G,{G_1}} \right\} = \left\{ {A_4^\varepsilon (q){,^3}{D_4}\left( {{q_1}} \right)} \right\}\), where (q − ϵ1)₅ ≠ 5 and q₁ > 2(7) \({A_4^\varepsilon (q)}\) and G₁ ∈ {B₃(q₁), C₃(q₁), D₄(q₁)}.

中文翻译：

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关于具有一致素数图的不同特征场上的小尺寸有限简单线性和Group群

假设G是一个有限群，π（G）是其阶数的主除数的集合，而ω（G）是其元素的阶数的集合。用下面的邻接关系的图被限定在π（ģ）：不同顶点ř和小号从π（ģ）相邻当且仅当RS＆Element; ω（ģ）。该图称为Gruenberg-Kegel图或G的素图，并用GK（G）。在来自“ Kourovka笔记本”的AV Vasil'ev的问题16.26中，要求用相同的Gruenberg-Kegel图描述所有成对的非同构简单非阿贝尔群。M. Hagie和MA Zvezdina在其中一个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。作者针对具有相同特征的字段上的有限个Lie类型简单组解决了这个问题。在本文中，我们证明以下定理。定理。令\（G = A_ {n-1} ^ \ pm \ left（q \ right）\），其中n∈{3，4，5，6}，令G ₁为Lie型的一个有限简单群与G非同构的q ₁阶场，其中q= p ^˚F，\（{Q_1} = P_1 ^ {{F_1}} \） ，和p和₁是不同的素数。如果图GK（G）和GK（G ₁）重合，则以下陈述之一成立（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）}，其中（q − 1）₃ ≠3，q + 1≠2 ^k和q ₁ > 2（5）\（\ left \ {G，G_ {1} \ right \} = \ left \ {A_ {4} ^ {\ varepsilon}（q），A_ {4} ^ {\ varepsilon_ {1} } \ left（q_ {1} \ right）\ right \} \），其中qq ₁是奇数（6）\（\ left \ {{G，{G_1}} \ right \} = \ left \ {{A_4 ^ \ varepsilon（q）{，^ 3} {D_4} \ left（{{q_1}} \ right）} \ right \} \），其中（q − ϵ1）₅ ≠5和q ₁ > 2（7）\（{A_4 ^ \ varepsilon（Q）} \）和G ^ ₁ ∈{乙₃（q ₁），Ç ₃（q ₁），d ₄（q ₁）}。