Page 303 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 303
GRAPHS AND GROUPS, GEOMETRIES AND GAP - G2G2 (MS-7)
[8] A. Valyuzhenich, Minimum supports of eigenfunctions of Hamming graphs, Discrete
Mathematics 340(5) (2017) 1064–1068.
[9] A. Valyuzhenich, Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph,
Discrete Mathematics 344(3) (2021) 112228.
[10] A. Valyuzhenich, K. Vorob’ev, Minimum supports of functions on the Hamming graphs
with spectral constraints, Discrete Mathematics 342(5) (2019) 1351–1360.
[11] K. V. Vorobev, D. S. Krotov, Bounds for the size of a minimal 1-perfect bitrade in a
Hamming graph, Journal of Applied and Industrial Mathematics 9(1) (2015) 141–146,
translated from Discrete Analysis and Operations Research 21(6) (2014) 3–10.
[12] K. Vorob’ev, I. Mogilnykh, A. Valyuzhenich, Minimum supports of eigenfunctions of
Johnson graphs, Discrete Mathematics 341(8) (2018) 2151–2158.
Closures of solvable permutation groups
Andrey V. Vasil’ev, vdr256@gmail.com
Sobolev Institute of Mathematics, Novosibirsk, Russian Federation
Coauthors: Eamonn A. O’Brien, Ilia Ponomarenko, Evgeny Vdovin
Let m be a positive integer and let Ω be a finite set. The m-closure G(m) 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. Wielandt [5] showed that
G(1) ≥ G(2) ≥ · · · ≥ G(m) = G(m+1) = · · · = G, (1)
for some m < |Ω|. (Since the stabilizer in G of all but one point is always trivial, G(n−1) = G
where n = |Ω|.) In this sense, the m-closure can be considered as a natural approximation
of G. It was shown by Praeger and Saxl [2] that for m ≥ 6, the m-closure G(m) of a primitive
permutation group G has the same socle as G. Furthermore, they classified explicitly primitive
groups G and H with different socles having the same m-orbits for m ≤ 5. Unfortunately, their
results say very little about closures of solvable permutation groups. The main goal of this talk
is to present the results of [1], where we study such closures.
The 1-closure of G is the direct product of symmetric groups Sym(∆), where ∆ runs over
the orbits of G. Thus the 1-closure of a solvable group is solvable if and only if each of its
orbits has cardinality at most 4. The case of 2-closure is more interesting. The 2-closure of
every (solvable) 2-transitive group G ≤ Sym(Ω) is Sym(Ω); other examples of solvable G and
nonsolvable G(2) appear in [4]. But, as shown by Wielandt [5], each of the classes of finite
p-groups and groups of odd order is closed with respect to taking the 2-closure. Currently, no
characterization of solvable groups having solvable 2-closure is known.
Seress [3] observed that if G is a primitive solvable group, then G(5) = G; so the 5-closure
of a primitive solvable group is solvable. Our main result is the following stronger statement.
Theorem 1. The 3-closure of a solvable permutation group is solvable.
The corollary below is an immediate consequence of Theorem 1 and the chain of inclu-
sions (1).
301
[8] A. Valyuzhenich, Minimum supports of eigenfunctions of Hamming graphs, Discrete
Mathematics 340(5) (2017) 1064–1068.
[9] A. Valyuzhenich, Eigenfunctions and minimum 1-perfect bitrades in the Hamming graph,
Discrete Mathematics 344(3) (2021) 112228.
[10] A. Valyuzhenich, K. Vorob’ev, Minimum supports of functions on the Hamming graphs
with spectral constraints, Discrete Mathematics 342(5) (2019) 1351–1360.
[11] K. V. Vorobev, D. S. Krotov, Bounds for the size of a minimal 1-perfect bitrade in a
Hamming graph, Journal of Applied and Industrial Mathematics 9(1) (2015) 141–146,
translated from Discrete Analysis and Operations Research 21(6) (2014) 3–10.
[12] K. Vorob’ev, I. Mogilnykh, A. Valyuzhenich, Minimum supports of eigenfunctions of
Johnson graphs, Discrete Mathematics 341(8) (2018) 2151–2158.
Closures of solvable permutation groups
Andrey V. Vasil’ev, vdr256@gmail.com
Sobolev Institute of Mathematics, Novosibirsk, Russian Federation
Coauthors: Eamonn A. O’Brien, Ilia Ponomarenko, Evgeny Vdovin
Let m be a positive integer and let Ω be a finite set. The m-closure G(m) 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. Wielandt [5] showed that
G(1) ≥ G(2) ≥ · · · ≥ G(m) = G(m+1) = · · · = G, (1)
for some m < |Ω|. (Since the stabilizer in G of all but one point is always trivial, G(n−1) = G
where n = |Ω|.) In this sense, the m-closure can be considered as a natural approximation
of G. It was shown by Praeger and Saxl [2] that for m ≥ 6, the m-closure G(m) of a primitive
permutation group G has the same socle as G. Furthermore, they classified explicitly primitive
groups G and H with different socles having the same m-orbits for m ≤ 5. Unfortunately, their
results say very little about closures of solvable permutation groups. The main goal of this talk
is to present the results of [1], where we study such closures.
The 1-closure of G is the direct product of symmetric groups Sym(∆), where ∆ runs over
the orbits of G. Thus the 1-closure of a solvable group is solvable if and only if each of its
orbits has cardinality at most 4. The case of 2-closure is more interesting. The 2-closure of
every (solvable) 2-transitive group G ≤ Sym(Ω) is Sym(Ω); other examples of solvable G and
nonsolvable G(2) appear in [4]. But, as shown by Wielandt [5], each of the classes of finite
p-groups and groups of odd order is closed with respect to taking the 2-closure. Currently, no
characterization of solvable groups having solvable 2-closure is known.
Seress [3] observed that if G is a primitive solvable group, then G(5) = G; so the 5-closure
of a primitive solvable group is solvable. Our main result is the following stronger statement.
Theorem 1. The 3-closure of a solvable permutation group is solvable.
The corollary below is an immediate consequence of Theorem 1 and the chain of inclu-
sions (1).
301
