TY - JOUR

T1 - The 3-closure of a solvable permutation group is solvable

AU - O'Brien, E. A.

AU - Ponomarenko, I.

AU - Vasil'ev, A. V.

AU - Vdovin, E.

N1 - Funding Information: O'Brien was supported by the Marsden Fund of New Zealand grant UOA 107 ; Ponomarenko and Vasil'ev were supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation ; Vdovin was supported by the RFBR grant No. 18-01-00752 . We thank the referee for helpful feedback. Publisher Copyright: © 2021 Elsevier Inc.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - Let m be a positive integer and let Ω be a finite set. The m-closure of G≤Sym(Ω) is the largest permutation group on Ω having the same orbits as G in its induced action on the Cartesian product Ωm. The 1-closure and 2-closure of a solvable permutation group need not be solvable. We prove that the m-closure of a solvable permutation group is always solvable for m≥3.

AB - Let m be a positive integer and let Ω be a finite set. The m-closure of G≤Sym(Ω) is the largest permutation group on Ω having the same orbits as G in its induced action on the Cartesian product Ωm. The 1-closure and 2-closure of a solvable permutation group need not be solvable. We prove that the m-closure of a solvable permutation group is always solvable for m≥3.

KW - Finite groups

KW - m-closure

KW - Permutation groups

UR - http://www.scopus.com/inward/record.url?scp=85110998304&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2021.07.002

DO - 10.1016/j.jalgebra.2021.07.002

M3 - Article

AN - SCOPUS:85110998304

VL - 607

SP - 618

EP - 637

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -