On Some arithmetic applications to the theory of symmetric groups
https://doi.org/10.22405/2226-8383-2023-24-4-252-263
Abstract
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 𝑆_𝑛 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 𝑆_𝑘 extends to an alternating group 𝐴_𝑘 for odd 𝑘.
The second part of the work examines some applications of prime number theory to cyclic subgroups of the symmetric group 𝑆_𝑛. 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 symmetric group 𝑆_𝑛.
About the Authors
Urusbi Mukhamedovich Pachev
Berbekov Kabardino–Balkarian State University; North–Caucasus Federal University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Rezuan Auesovich Dokhov
North–Caucasus Federal University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
Azamat Khasanovich Kodzokov
Berbekov Kabardino–Balkarian State University
Russian Federation
Russian Federation
Marina Sefovna Nirova
Berbekov Kabardino–Balkarian State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
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Review
For citations:
Pachev U.M., Dokhov R.A., Kodzokov A.Kh., Nirova M.S. On Some arithmetic applications to the theory of symmetric groups. Chebyshevskii Sbornik. 2023;24(4):252-263. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-252-263
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