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Finite totally k-closed groups1. / Churikov, Dmitry; Praeger, Cheryl E.
в: Trudy Instituta Matematiki i Mekhaniki UrO RAN, Том 27, № 1, 20, 2021, стр. 240-245.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - Finite totally k-closed groups1
AU - Churikov, Dmitry
AU - Praeger, Cheryl E.
N1 - Publisher Copyright: © 2021 Krasovskii Institute of Mathematics and Mechanics. All Rights Reserved.
PY - 2021
Y1 - 2021
N2 - For a positive integer k, a group G is said to be totally k-closed if in each of its faithful permutation representations, say on a set , G is the largest subgroup of Sym() which leaves invariant each of the G-orbits in the induced action on × × = k. We prove that every finite abelian group G is totally (n(G) + 1)- closed, but is not totally n(G)-closed, where n(G) is the number of invariant factors in the invariant factor decomposition of G. In particular, we prove that for each k ≥ 2 and each prime p, there are infinitely many finite abelian p-groups which are totally k-closed but not totally (k - 1)-closed. This result in the special case k = 2 is due to Abdollahi and Arezoomand. We pose several open questions about total k-closure.
AB - For a positive integer k, a group G is said to be totally k-closed if in each of its faithful permutation representations, say on a set , G is the largest subgroup of Sym() which leaves invariant each of the G-orbits in the induced action on × × = k. We prove that every finite abelian group G is totally (n(G) + 1)- closed, but is not totally n(G)-closed, where n(G) is the number of invariant factors in the invariant factor decomposition of G. In particular, we prove that for each k ≥ 2 and each prime p, there are infinitely many finite abelian p-groups which are totally k-closed but not totally (k - 1)-closed. This result in the special case k = 2 is due to Abdollahi and Arezoomand. We pose several open questions about total k-closure.
KW - K-closure
KW - Permutation group
KW - Totally k-closed group
UR - http://www.scopus.com/inward/record.url?scp=85112033511&partnerID=8YFLogxK
UR - https://www.elibrary.ru/item.asp?id=44827408
U2 - 10.21538/0134-4889-2021-27-1-240-245
DO - 10.21538/0134-4889-2021-27-1-240-245
M3 - Article
AN - SCOPUS:85112033511
VL - 27
SP - 240
EP - 245
JO - Trudy Instituta Matematiki i Mekhaniki UrO RAN
JF - Trudy Instituta Matematiki i Mekhaniki UrO RAN
SN - 0134-4889
IS - 1
M1 - 20
ER -
ID: 34193034