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

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). In a series of papers, we describe the coincidence conditions for the prime graphs of nonisomorphic simple groups. This issue is connected with Vasil’ev’s Question 16.26 in the Kourovka Notebook about the number of nonisomorphic simple groups with the same prime graph. Earlier the author derived necessary and sufficient conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type over fields of orders q and q1, respectively, with the same characteristic. Let G and G1 be two nonisomorphic finite simple groups of Lie type over fields of orders q and q1, respectively, with different characteristics. The author also obtained necessary conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type. In the present paper the latter result is refined in the case where G is a simple linear group of sufficiently high Lie rank over a field of order q. If G is a simple linear group of sufficiently high Lie rank, then we prove that the prime graphs of G and G1 may coincide only in one of the nineteen cases. As corollaries of the main result, we obtain constraints (under some additional conditions) on the possible number of simple groups whose prime graph is the same as the prime graph of a simple linear group.