On Some arithmetic applications to the theory of symmetric groups

U. M. Pachev ; R. A. Dokhov ; A. H. Kodzokov ; M. S. Nirova

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 252-263.

Résumé

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 $k$, $\varphi(k)$, and $k \varphi(k)$, where $k \leq n$, $\varphi$ – Euler function, those representations jf grups $\left( \mathbb{Z} / k\mathbb{Z}, + \right)$, $\left( \mathbb{Z} / k\mathbb{Z} \right)^{*}$ 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 $\mathbb{Z} / k\mathbb{Z}$ residue class rings, where $\gcd\left(a, k\right) = 1$. In addition, the result concerning the isomorphic embedding of a group $\left( \mathbb{Z} / k\mathbb{Z} \right)^{*}$ in to a group $\left( \mathbb{Z} / k\mathbb{Z} \right)^{*}$ 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 summetric group $S_{n}$.

Détail

Keywords: symmetric group, subgroup order, modulo congruence, Euler function, substitution sign, quadratic residnes, permutation polynomial, prime divisor of cyclic subgroup order.

Bibliographie

[1] Borevich Z. I., Zhafarevich I. R., Number theory, Nauka, Moscow, Russia, 1985, 510 pp. | MR

[2] Bertran, Journ. de l'Ecole Polyt., 1845

[3] Landau E., “Über die Maximalordnung der Permutation gegeben Grads”, Archiv der Math. Und Phus., Ser. 3, 5 (1903), 92–103

[4] Shah S., “An Ineguality for the Arithmetical Function $g(x)$. I”, Indian Math. Soc., 3 (1939), 316–318 | MR | Zbl

[5] Massian I., “Majoration explite de l'ordre maximum d'un element du group symetrique”, Ann. Fac. Shi. Toulouse Math., 5:6 (1984) ; 3–4, 269–281 | MR

[6] Szalay M., “On the Maximal Order in $S_{n}$ and $S_{n}^{*}$”, Acta Arith., 37 (1980), 321–331 | DOI | MR | Zbl

[7] Nicolas I. L., “Ordre maximal d'un element du group des permutations et highly composite numbers”, Bull. Soc. Math. Franke, 97 (1969), 129–191 | DOI | MR | Zbl

[8] Nathanson M. B., “On the Greatest Order of an Element of the Symmetric Group”, this Monthly, 79 (1972), 500–501 | MR | Zbl

[9] Miller W., “The Maximum Order of an Element of a Finite Symmetric Group”, The American Mathematical Montly, 94:6 (1987), 497–506 | DOI | MR | Zbl

[10] Pachev U. M., Shokuev V. N., “On thr application of the theory of congruence to the study of subgroups in symmetric groups”, Jzw. KBNTs RAS, 2001, no. 1(6), 68–69

[11] Dokhov R. A., Pachev U. M., “On the maximum number of primedivisors of orders of cyclic subgroups in $a$ symmetris group”, Procceedings of the XXII International Conference “Algebra, number theory, discrete geometry and multiscale modeling: modern problems, applications and prolems of history” (Tula, 2023), 173–174

[12] Vinogradov I. M., Fundavental of Number Theoury, Nauka, M., 1981, 168 pp. | MR

[13] Kargapolov M. I., Merzlyakov Yu. I, Fundamentals of group theory, Nauka, M., 1982, 288 pp. | MR

[14] Zolotareff G., “Nouvelle demonstration de la loi de reciprocite de Legendre”, Nouv. Ann. Math. (2), 11 (1872), 354–362

[15] Gashkov S. B, Modern Elementary Algebra in Problems and Exercises, MCNMO, M., 2006, 328 pp.

[16] Gecke E., Lectures on the theory of algebraic numbers, M.–L., 1940, 260 pp.

[17] Trost E, Prime numbers, GIFML, M., 1959, 136 pp.

[18] Buhshtab A. A., Number theory, M., 1966

[19] Rosser B., “The $n$-th Prime is greater than $n \log n$”, Proc. London. Math. Soc., 45 (1938), 21–44 | MR | Zbl

[20] Krätzel E., Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1981 | MR | Zbl

[21] Lidl R., Niederreiter G., Finite Fields, v. 2, Mir, M., 1988, 437–820

[22] Pachev U. M., Selected chapter of number theory, Nalchik, 2016, 186 pp.