The 3-closure of a solvable permutation group is solvable

The closures of wreath products in product action

“…Theorem 1.2 gives a natural generalization of the two mentioned results from [4] and [8]. The exceptional case m = 2 and K is 2-transitive cannot be avoided, see example in (2).…”

“…When the vertex set of this graph is of the form (5), the edge set is an orbit of Sym(Γ) ↑ Sym(∆). It almost immediately follows that (8) (Sym(Γ) ↑ Sym(∆)) (2) = Sym(Γ) ↑ Sym(∆).…”

“…The inclusion (K ↑ L) (m) ≤ K (m) ↑ L (m) was first proved in [4,Proposition 3.1] for m = 2 and non 2-transitive K (cf. ( 2)), and then recently in [8,Theorem 3.3] for m ≥ 3, primitive K ↑ L, and technical assumptions on the degrees of K and L.…”

“…In general, studying the m-closure for m ≥ 2 is a nontrivial problem both from theoretical and computational point of view, see, e.g., [4,[7][8][9][10]12]. Usual approach here is a reduction via the direct or wreath products to smaller permutation groups.…”

“…The number k m defined in Theorem 1.1 is bounded from below by the number | Orb m (Sym(n))|, where n is the degree of K, which is equal to the number of ordered partitions of a set of cardinality m (see, e.g., [8,Example 2.1]). In particular,…”