The graph of atomic divisors and recognition of finite simple groups

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).

The time complexity of some algorithms for generating the spectra of finite simple groups

“…The results of this paper complement the main theorem of [5], which is concerned with the following problem. Given a finite set M of natural numbers, determine whether there exists a finite simple group G whose spectrum coincide with the set of divisors of elements of M. It is proved in [5] that there is a polynomial-time algorithm which returns a unique candidate for G (more precisely, its parameters in the sense described above) or says that there is no such group.…”

“…Given a finite set M of natural numbers, determine whether there exists a finite simple group G whose spectrum coincide with the set of divisors of elements of M. It is proved in [5] that there is a polynomial-time algorithm which returns a unique candidate for G (more precisely, its parameters in the sense described above) or says that there is no such group. This candidate G satisfies the following conditions: M is a subset of ω(G) and if H is a finite simple group whose spectrum differs from the spectrum of G, then the set of divisors of elements of M is not the spectrum of H. Thus, to complete the task it remains to verify whether ω(G) is equal to the set of divisors of elements of M, or equivalently, whether µ(G) is a subset of M. Hence the following statements are corollaries of the main result of [5] and Theorems 1, 2 and 3.…”