Closures of permutation groups with restricted nonabelian composition factors

Given a permutation group G on a finite set Ω, let G⁽ᵏ⁾ denote the k-closure of G, that is, the largest permutation group on Ω having the same orbits in the induced action on Ωᵏ as G. Recall that a group is Alt(d)-free if it does not contain a section isomorphic to the alternating group of degree d. Motivated by some problems in computational group theory, we prove that the k-closure of an Alt(d)-free group is again Alt(d)-free for k ≥ 4 and d ≥ 25.