On Constructive Recognition of Finite Simple Groups by Element Orders

Abstract

The set of element orders of a finite group G is called the spectrum and is denoted by ω(G), and groups with the same spectrum are said to be isospectral. The following question seems to be natural: If M is a set of positive integers, does then a finite group G with ω(G) = M exist, and if so, can we describe all such groups? The paper is concerned with an algorithmic aspect of this problem in the special case where G is simple. The last restriction is justified by the following reasons. Obviously, there exist nonisomorphic isospectral finite groups; moreover, if G is a finite group with nontrivial soluble radical, then we can construct infinitely many pairwise nonisomorphic finite groups isospectral to G (see, e.g., [1]). On the other hand, in general, the set of groups isospectral to a finite non-Abelian simple group G is finite and consists of groups closely related to G (see, e.g., [2,3]). Thus if the name (according to CFSG) of a simple group G with ω(G) = M is determined, then normally a complete list of groups enjoying this property can be created.Note that the spectrum of a group G is closed under taking divisors; i.e., if n ∈ ω(G) and d divides n then d ∈ ω(G). In particular, ω(G) is uniquely determined by its subset μ(G) of elements maximal w.r.t. divisibility. Let ω(M) and μ(M) stand for respectively the set of all divisors of elements of M and the set of all elements of M maximal w.r.t. divisibility. If we require ω(G) = ω(M) instead of ω(G) = M, then we obtain a more natural problem whose solution obviously implies a solution to the original problem.

The graph of atomic divisors and recognition of finite simple groups

“…Finally, let G = O ε 2n (q). The description of spectra of these groups (see, for example, [6,Corollaries 4,8,9]) imply that Φ * i (q) for i 2 and Φ * 1 (q) t ′ divide some vertices of AD(G). As before, denote by v i the vertex of AD(G) divisible by Φ * i (q) for i 2 and the vertex divisible by (Φ * 1 (q)) t ′ for i = 1.…”

“…The main tool of the proof of Theorem 1 is a notion of the graph of atomic divisors (briefly, the AD-graph) of a set M of positive integers (introduced as M-graph in [9] where our present result was announced). In the case when M = µ(G) for a group G, this graph (denoted as AD(G)) shares some of substantial features with the so-called prime graph GK(G) (defined in [24]) and, as the latter one, reflects essential properties of G itself.…”