On finite simple classical groups 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. 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 G₁ be two nonisomorphic finite simple groups of Lie type over fields of orders q and q₁, 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 G₁ may coincide only in one of three cases. It is also proved that, if G = A₁(q) and G₁ is a classical group, then the prime graphs of the groups G and G₁ coincide only if {G, G₁} is equal to {A₁(9), A₁(4)}, {A₁(9), A₁(5)}, {A₁(7), A₁(8)}, or {A₁(49),²A₃(3)}.