Closures of permutation groups with restricted nonabelian composition factors

Given a permutation group G on a finite set, let G^ (k) denote the k-closure of G, that is, the largest permutation group on having the same orbits in the induced action on ᵏ 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.