Order-divisibility bijections onto finite simple groups and Kourovka Problem 21.2

Problem 21.2 of the Kourovka Notebook, posed by M. Amiri, asks the following. Let S be a finite simple group and let G be a finite group admitting a bijection f : G → S such that |x| divides |f(x)| for every x ∈ G. Must G be simple? Suppose that some union π of connected components of the prime graph of S, with 2 ∉ π, satisfies an explicit arithmetic condition. We show that then the Fitting subgroup of G is trivial and G is almost simple, and that either of two further checkable conditions makes G simple. The proofs combine counting inequalities forced by the bijection with Thompson’s theorem on fixed-point-free automorphisms of prime order. As applications we obtain an affirmative answer for PSL(2, q) with q ≥ 5 prime, for every nonabelian simple group of order less than |A10| = 1 814 400, for every sporadic simple group except possibly Fi22, and for 115 of the 154 nonabelian simple groups whose character tables are in version 1.3.7 of the GAP Character Table Library. The last three results are computer-assisted and use the classification of finite simple groups.