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. We define a graph on π(G) with the following adjacency relation: 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). Let G and G 1 be two nonisomorphic finite simple groups of Lie type over fields of orders q and q 1, respectively, with different characteristics. It is proved that, if G is a classical group of a sufficiently high Lie rank, then the prime graphs of the groups G and G 1 may coincide only in one of three cases. It is also proved that, if G = A 1(q) and G 1 is a classical group, then the prime graphs of the groups G and G 1 coincide only if {G, G 1} is equal to {A 1(9), A 1(4)}, {A 1(9), A 1(5)}, {A 1(7), A 1(8)}, or {A 1(49),2 A 3(3)}.
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Original Russian Text © M.R. Zinov’eva, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 3.
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Zinov’eva, M.R. On finite simple classical groups over fields of different characteristics with coinciding prime graphs. Proc. Steklov Inst. Math. 297 (Suppl 1), 223–239 (2017). https://doi.org/10.1134/S0081543817050248
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DOI: https://doi.org/10.1134/S0081543817050248