On Some arithmetic applications to the theory of symmetric groups

The work is devoted to some arithmetic applications to the theory of symmetric groups. Using the properties of congruences and classes of residues from number theory, the existence in the symmetric group Sn of degree n of cyclic, Abelian and non-Abelian subgroups respectively, of orders is establisn...

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Главные авторы:	Pachev, U. M., Пачев, У. М., Dokhov, R. A., Дохов, Р. А.
Формат:	Статья
Язык:	Russian
Опубликовано:	State Lev Tolstoy Pedagogical University 2024
Темы:	Euler function Symmetric group Modulo congruence Permutation polynomial Prime divisor of cyclic subgroup order Subgroup order Quadratic residnes Substitution sign
Online-ссылка:	https://dspace.ncfu.ru/handle/123456789/29283
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spelling	ir-123456789-292832024-11-28T13:13:15Z On Some arithmetic applications to the theory of symmetric groups О некоторых арифметических применениях к теории симметрических групп Pachev, U. M. Пачев, У. М. Dokhov, R. A. Дохов, Р. А. Euler function Symmetric group Modulo congruence Permutation polynomial Prime divisor of cyclic subgroup order Subgroup order Quadratic residnes Substitution sign The work is devoted to some arithmetic applications to the theory of symmetric groups. Using the properties of congruences and classes of residues from number theory, the existence in the symmetric group Sn of degree n of cyclic, Abelian and non-Abelian subgroups respectively, of orders is establisned k, φ(k), and kφ(k), where k ≤ n, φ – Euler function, those representations jf grups (Z/kZ, +), (Z/kZ)* and theorem product in the form of degree substitutions k. In this case isomorphic embeddings of these groups are constructed following the proof of Cayley’s theorem, but along with this, a linear binomial is used Z/kZ residue class rings, where gcd (a, k) = 1. In addition, the result concerning the isomorphic embedding of a group (Z/kZ)* in to a group (Z/kZ)* in to a group Sk extends to an alternating group Ak for odd k. The second part of the work examines some applications of prime number theory to cyclic subgroups of the symmetric group Sn. In particular, applying the Euler-Maclaurin summation formula and bounds for the k in prime, a lower bound for maximum number of prime divisors of cyclic orders in the summetric group Sn. 2024-11-28T13:12:08Z 2024-11-28T13:12:08Z 2023 Статья Pachev U.M., Dokhov R.A., Kodzokov A.K., Nirova M.S. On Some arithmetic applications to the theory of symmetric groups // Chebyshevskii Sbornik. - 2023. - 24 (4). - pp. 252 - 263. - DOI: 10.22405/2226-8383-2023-24-4-252-263 https://dspace.ncfu.ru/handle/123456789/29283 ru Chebyshevskii Sbornik application/pdf State Lev Tolstoy Pedagogical University
institution	СКФУ
collection	Репозиторий
language	Russian
topic	Euler function Symmetric group Modulo congruence Permutation polynomial Prime divisor of cyclic subgroup order Subgroup order Quadratic residnes Substitution sign
spellingShingle	Euler function Symmetric group Modulo congruence Permutation polynomial Prime divisor of cyclic subgroup order Subgroup order Quadratic residnes Substitution sign Pachev, U. M. Пачев, У. М. Dokhov, R. A. Дохов, Р. А. On Some arithmetic applications to the theory of symmetric groups
description	The work is devoted to some arithmetic applications to the theory of symmetric groups. Using the properties of congruences and classes of residues from number theory, the existence in the symmetric group Sn of degree n of cyclic, Abelian and non-Abelian subgroups respectively, of orders is establisned k, φ(k), and kφ(k), where k ≤ n, φ – Euler function, those representations jf grups (Z/kZ, +), (Z/kZ)* and theorem product in the form of degree substitutions k. In this case isomorphic embeddings of these groups are constructed following the proof of Cayley’s theorem, but along with this, a linear binomial is used Z/kZ residue class rings, where gcd (a, k) = 1. In addition, the result concerning the isomorphic embedding of a group (Z/kZ)* in to a group (Z/kZ)* in to a group Sk extends to an alternating group Ak for odd k. The second part of the work examines some applications of prime number theory to cyclic subgroups of the symmetric group Sn. In particular, applying the Euler-Maclaurin summation formula and bounds for the k in prime, a lower bound for maximum number of prime divisors of cyclic orders in the summetric group Sn.
format	Статья
author	Pachev, U. M. Пачев, У. М. Dokhov, R. A. Дохов, Р. А.
author_facet	Pachev, U. M. Пачев, У. М. Dokhov, R. A. Дохов, Р. А.
author_sort	Pachev, U. M.
title	On Some arithmetic applications to the theory of symmetric groups
title_short	On Some arithmetic applications to the theory of symmetric groups
title_full	On Some arithmetic applications to the theory of symmetric groups
title_fullStr	On Some arithmetic applications to the theory of symmetric groups
title_full_unstemmed	On Some arithmetic applications to the theory of symmetric groups
title_sort	on some arithmetic applications to the theory of symmetric groups
publisher	State Lev Tolstoy Pedagogical University
publishDate	2024
url	https://dspace.ncfu.ru/handle/123456789/29283
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On Some arithmetic applications to the theory of symmetric groups

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