Two-closures of supersolvable permutation groups in polynomial time

On finite totally 2-closed groups

“…In 1969, Wielandt initiated the study of 2-closures of permutation groups to present a unified treatment of finite and infinite permutation groups, based on invariant relations and invariant functions [22]. After Wielandt's pioneering work, there was some progress on the subject achieved mostly in the case of primitive groups [12,13,16,19,20,25] and the 2-closure was used as a tool in studying the graph isomorphism problem [9,17,18]; the isomorphism problem for Schurian coherent configurations [10,21]; and in the study of automorphisms of vertex transitive graphs [7,23,24]. The latter of these led to the formulation of the Polycirculant conjecture [5], which remains open, and has garnered much recent attention [2].…”

Two-closure of rank 3 groups in polynomial time

“…Such a coherent configuration is essentially a colored graph, and the full automorphism group of this graph is the 2-closure of the original group. This point of view initiated by Ponomarenko in [28] led to a series of results computing 2-closures of permutation groups from various classes, such as, groups of odd order [13], nilpotent groups [28], 3 2 -transitive groups [36] and supersolvable groups [26]. Our contribution can be thought as a continuation of this line of research.…”

The closures of wreath products in product action

“…Let Ω [m] be the set of all ordered partitions Π of Ω such that |Π| ≤ m. For a group G ≤ Sym(Ω), we denote by G [m] the largest permutation group on Ω having the same orbits as G in its induced action on Ω [m] . Obviously, (9) Sym(Ω) = G [1] ≥ G [2]…”

“…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.…”