Quasirecognition by prime graph of the simple group 2 G 2(q)

Abstract

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ( 2 G 2 (q)), where q = 3 2n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2 G 2 (q). We infer that if G is a finite group satisfying |G| = | 2 G 2 (q)| and Γ(G) = Γ( 2 G 2 (q)) then G ∼ = 2 G 2 (q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.

Recognition of finite groups by the prime graph

“…If (r, a) = (2,19) or (3,7) then, by Lemma 5, N.P contains an element of order ra whenever N = 1. Hence N = 1 and L 3 (7)…”

Quasirecognition by prime graph of $C_{n}(4) $, where $ n\geq17 $ is odd

“…The prime graph (or Gruenberg-Kegel graph) Γ(G) of G is the graph with vertex set π(G) where two distinct vertices p and q are adjacent by an edge (we write (p, q) ∈ Γ(G)) if p.q ∈ ω(G) and we denote by s(G) the number of connected components of Γ(G). A finite non-abelian simple group G is quasirecognizable by its prime graph, if each finite group P with Γ(P ) = Γ(G) has a unique non-abelian composition factor isomorphic to G [5]. The most recent lists of finite simple groups that are quasirecognizable by prime graph are presented in [2], [4], [6], [7] and [8].…”

On the composition factors of a group with the same prime graph as B n (5)

“…A finite group G is called recognizable by prime graph if Γ(H) = Γ(G) implies that H ∼ = G. A nonabelian simple group P is called quasirecognizable by prime graph if every finite group whose prime graph equals Γ(P ) has a unique nonabelian composition factor isomorphic to P (see [11]). Obviously, recognition (quasirecognition) by prime graph implies recognition (quasirecognition) by spectrum, but the converse is not true in general.…”

“…It is proved that if q = 3 2n+1 (n > 0), then the simple group 2 G 2 (q) is recognizable by its prime graph [11], [27]. A group G is called a CIT group if G is of even order and the centralizer in G of any involution is a 2-group.…”