Characterization of Groups E6(3) and 2E6(3) by Gruenberg Kegel Graph

Abstract

The Gruenberg Kegel graph (or the prime graph) Γ(G) of a nite group G is de ned as follows. The vertex set of Γ(G) is the set of all prime divisors of the order of G. Two distinct primes r and s regarded as vertices are adjacent in Γ(G) if and only if there exists an element of order rs in G. Suppose that L =≅ E6(3) or L ≅= 2E6(3). We prove that if G is a nite group such that Γ(G) = Γ(L), then G ≅= L. © 2021 Khramova A.P., Maslova N.V., Panshin V.V., Staroletov A.M. The work is supported by the Mathematical Center in Akademgorodok under the agreement 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation.