On Finite Simple Linear and Unitary Groups of Small Size over Fields of Different Characteristics with Coinciding Prime Graphs

Abstract

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)}.