The connection between the linear Diophantine equations solutions under the actions of the symmetric group and the automoprhism group of the integers
https://doi.org/10.22405/2226-8383-2025-26-5-259-279
Abstract
This article shows that there is a direct connection between the actions of the symmetric
group 𝑆𝑛 on the set of linear Diophantine equations and on the set of their solutions.
Thus, it was found that for coefficients rearrangement in a linear Diophantine equation the
coordinates of its general solution vector are rearranged in the same order, and for variables rearrangement we get the reverse order of the vector coordinates.
There is a similar connection between the actions of the group of the automorphisms of the
integers on the set of linear Diophantine equations and their solutions: if on changes the signs of some coefficients in the equation, then the signs of the corresponding coordinates of the general solution vector are changed too.
We also obtained results on the connection of different actions of the symmetric group on
the set of linear Diophantine equations. For example, rearrangement of the equation coefficients is equivalent to rearrangement of its variables in the reverse order.
Established connections between the actions make it possible to quickly find solutions of an
entire class of linear diophantine equations from a solution of only one of its elements. In turn, studying the actions of other groups on the given set and the connections generated by them would expand this class.
About the Authors
Ivan Sergeevich Chistov
Moscow Pedagogical State University
Russian Federation
Russian Federation
student
Liliya Mikhailovna Tsybulya
Moscow Pedagogical State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
References
1. Kostrikin, A. I. 2004, “Introduction to algebra. Part III. Basic structures: textbook for universities”, M.: FIZMATLIT, Third Edition, 272 p. (In Russian).
2. Vinberg, E. B. 2001, “Algebra course”, M.: Faktorial Press, Second Edition, 544 p. (In Russian).
3. Buhshtab, A. A. 1966, “Number theory”, M.: Prosveshchenie, 384 p. (In Russian).
4. Serpinskij, V. 1961, “On solving equations in integers”, M.: Gosudarstvennoe izdatel’stvo fizikomatematicheskoj literatury, 88 p. (In Russian).
5. Nesterenko, Yu. V. 2008, “Number Theory: a textbook for university students”, M.: Izdatel’skij centr «Akademiya», 272 p. (In Russian).
6. Monahov, V. S. 2003, “Introduction to the theory of finite groups and their classes: а textbook”, Gomel’: UO «GGU im. F. Skoriny», 322 p. (In Russian).
7. Kaluzhnin, L. A., Sushchanskij V. I. 1985, “Transformations and permutations: translation from Ukrainian”, M.: Nauka. Glavnaya redakciya fiziko-matematicheskoj literatury, Second Edition, 160 p. (In Russian).
8. Aleksandrov, P. S. 2017, “Introduction to group theory”, M.: LENAND, 128 p. (In Russian).
9. Kurosh, A. G. 1968, “Higher Algebra Course”, M.: Nauka, Ninth Edition, 431 p. (In Russian).
10. Artin, Michael 1991, “Algebra”, Prentice Hall, 618 p.
11. Streng, G. 1980, “Linear algebra and its applications”, M: Mir, 454 p. (In Russian).
12. Artamonov, V. A., Salij V. N., Skornyakov L. A. and others under the general editorship of L. A. Skornyakov 1991, “General algebra”, M.: Nauka. Gl. red. fiz.-mat. lit., Volume 2, 480 p. (In Russian).
13. Fuks, L. 1977, “Infinite Abelian groups”, M.: Mir, Volume 2, 416 p. (In Russian).
14. Kostrikin, A. I. 2000, “Introduction to algebra. Part II. Linear algebra: a textbook for universities”, M.: Fiziko-matematicheskaya literatura, 368 p. (In Russian).
15. Chistov, I. S., Tsybulya, L. M. 2024, “Some dependencies of the linear Diophantine equations solutions under the actions of the symmetric group and the automoprhism group of the integers”, Algebra, number theory, discrete geometry and multiscale modeling: Modern problems, applications and problems of history: Proceedings of the XXIII International Conference devoted the 80-th anniversary of the birth of professor Aleksandr Ivanovich Galochkin and the 75-th
16. anniversary of the professor Vladimir Grigor’evich Chirskij. — Tula: Tula State Lev Tolstoy Pedagogical University, pp. 38 – 41. (In Russian).
Review
For citations:
Chistov I.S., Tsybulya L.M. The connection between the linear Diophantine equations solutions under the actions of the symmetric group and the automoprhism group of the integers. Chebyshevskii Sbornik. 2025;26(5):259-279. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-5-259-279
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