The graph of atomic divisors and recognition of finite simple groups

Abstract

Abstract The spectrum ω ( G ) of a finite group G is the set of orders of elements of G. We present a polynomial-time algorithm that, given a finite set M of positive integers, outputs either an empty set or a finite simple group G. In the former case, there is no finite simple group H with M = ω ( H ) , while in the latter case, M ⊆ ω ( G ) and M ≠ ω ( H ) for all finite simple groups H with ω ( H ) ≠ ω ( G ) .