Recognition of the finite almost simple groups PGL2(q) by their spectrum

For a finite group G, denote by ω(G) the spectrum of G, i.e., the set of orders of elements in G. This set is closed under divisibility and hence is uniquely determined by the subset ω(G) of elements in ω(G) which are maximal under the divisibility relation. A group G is said to be recognizable by ω(G) (for short, recognizable) if every finite group H with ω(H) = ω(G) is isomorphic to G. In other words, G is recognizable if h(G) = 1 where h(G) is the number of pairwise non-isomorphic groups H with ω(H) = ω(G). It is known that h(G) = ∞ for every group G that has a non-trivial soluble normal subgroup, and so the recognizability problem is interesting only for groups with trivial soluble radical, and first of all for simple and almost simple groups. The goal of this paper is to resolve the recognizability problem for the groups PGL2(q), i.e., to find h(PGL2(q)) for all q. © Walter de Gruyter 2007.