OD-Characterization of Some Projective Special Linear Groups Over the Binary Field and Their Automorphism Groups

Abstract

Listen

Translate article

The Gruenberg–Kegel graph GK(G) = (V G , E G ) of a finite group G is a simple graph with vertex set V G = π(G), the set of all primes dividing the order of G, and such that two distinct vertices p and q are joined by an edge, {p, q} ∈ E G , if G contains an element of order pq. The degree deg G (p) of a vertex p ∈ V G is the number of edges incident to p. In the case when π(G) = {p 1, p 2,…, p h } with p 1 < p 2 < … <p h , we consider the h-tuple D(G) = (deg G (p 1), deg G (p 2),…, deg G (p h )), which is called the degree pattern of G. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying condition (|H|, D(H)) = (|G|, D(G)). Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we prove that the simple groups L 10(2) and L 11(2) are OD-characterizable. It is also shown that automorphism groups Aut(L p (2)) and Aut(L p+1(2)), where 2 p − 1 is a Mersenne prime, are OD-characterizable. Finally, a list of finite (simple) groups which are presently known to be k-fold OD-characterizable, for certain values of k, is presented.