On constructive nilpotent groups
72 37
Keywords:
nilpotent group, unitriangular group of matrices over the ring of polynomials in one variable with integer coefficients, the center of the groupAbstract
In connection with the development of the theory of algorithms a study of the problems of computability of important classes of algebraic systems are currently relevant. Groups of unitriangular matrices over the ring are a classic representative of the class of nilpotent groups and have numerous applications both in group theory and in its applications. In this paper we investigate the questions of computability of nilpotent groups. The connection between the bases of the subgroup of the nilpotent torsion-free group and the quotient of this subgroup are found here. Sufficient condition of computability of nilpotent groups in terms of their subgroups and quotient by this subgroup are given. On the basis of these conditions, we exhibit find a large class of computable subgroups of the group of unitriangular matrices of degree 3 over the ring of polynomials in one variable with integer coefficients. In particular, it is proved that every Abelian subgroup of this group is computable. It has been established that any computable numbering of a non-Abelian subgroup of the group of all unitriangular matrices of degree 3 over the polynomial ring in one variable with integer coefficients induces in any computable numbering its maximal subgroups and quotient of it. We obtain a sufficient condition for computability of the quotient of a computable nilpotent group by its periodic part. We find a sufficient condition for computability of a nilpotent group, enriched by an additional predicate root extraction.References
References
[1] Mal’cev A. I. Recursive Abelian groups // Soviet Math. Dokl. – 1962. – №4 (46). – P. 1009—1012.
[2] Goncharov S. S., Ershov Yu. L. Constructive models. — Novosibirsk: Nauchnay kniga, 1996.
[3] Ershov Yu. L. Existence of constructivizations // Soviet Math. Dokl. – 1972. – №5 (204). – P. 1041—1044.
[4] Goncharov S. S., Molokov A. V., Romanovskij N. S. Nilpotent groups of finite algorithmic dimension // Siberian Math. J. – 1989. – №1 (30). – P. 82—88.
[5] Goncharov S. S., Drobotun B. N. Algorithmic dimension of nilpotent groups // Siberian Math. J. – 1989. – №2 (30). –
P. 52—60.
[6] Latkin I. V. Arithmetic hierarchy of torsion-free nilpotent groups // Algebra and Logic. – 1996. – №3 (35). – P. 308—313.
[7] Roman’kov V. A., Khisamiev N. G. Constuctive matrix and orderable groups // Algebra and Logic. – 2004. – №3 (43).
– P. 353—363.
[8] Roman’kov V. A., Khisamiev N. G. Constructible matrix groups // Algebra and Logic. – 2004. – №5 (43). – P. 603—613.
[9] Khisamiev N. G. On constructive nilpotent groups // Siberian Math. J. – 2007. – №1 (48). – P. 214—223.
[10] Khisamiev N. G. Positively related nilpotent groups // Math. Zh. – Almaty, 2007. – №2 (24). – P. 95—102.
[11] Khisamiev N. G. Torsion-free constructive nilpotent Rp groups // Siberian Math. J. – 2009. – №1 (50). – P. 222—230.
[12] Khisamiev N. G. Hierarchies of torsion-free Abelian groups // Algebra and Logic. – 1986. – №2 (25). – P. 205—226.
[13] Khisamiev N. G. On positive and constructive groups // Siberian Math. J. – 2012. – №5 (53). – P. 1133—1146.
[14] Nurizinov М. K., Tyulyubergenev R. K., Khisamiev N. G. Computable torsion-free nilpotent groups of finite dimension
// Siberian Math. J. – 2014. – №3 (55). – P. 580—591.
[15] Kargopolov M. I., Merzlyakov Yu. I. Fundamentals of the Theory of Groups. — Moscow: Nauka, 1996.
[16] Mal’cev A. I. Algorithms and Recursive Functions. — Moscow: Nauka, 1986.
[17] Fuchs L. Infinite Abelian groups, Mir, trans. From English, Moscow, 1974.
[1] Mal’cev A. I. Recursive Abelian groups // Soviet Math. Dokl. – 1962. – №4 (46). – P. 1009—1012.
[2] Goncharov S. S., Ershov Yu. L. Constructive models. — Novosibirsk: Nauchnay kniga, 1996.
[3] Ershov Yu. L. Existence of constructivizations // Soviet Math. Dokl. – 1972. – №5 (204). – P. 1041—1044.
[4] Goncharov S. S., Molokov A. V., Romanovskij N. S. Nilpotent groups of finite algorithmic dimension // Siberian Math. J. – 1989. – №1 (30). – P. 82—88.
[5] Goncharov S. S., Drobotun B. N. Algorithmic dimension of nilpotent groups // Siberian Math. J. – 1989. – №2 (30). –
P. 52—60.
[6] Latkin I. V. Arithmetic hierarchy of torsion-free nilpotent groups // Algebra and Logic. – 1996. – №3 (35). – P. 308—313.
[7] Roman’kov V. A., Khisamiev N. G. Constuctive matrix and orderable groups // Algebra and Logic. – 2004. – №3 (43).
– P. 353—363.
[8] Roman’kov V. A., Khisamiev N. G. Constructible matrix groups // Algebra and Logic. – 2004. – №5 (43). – P. 603—613.
[9] Khisamiev N. G. On constructive nilpotent groups // Siberian Math. J. – 2007. – №1 (48). – P. 214—223.
[10] Khisamiev N. G. Positively related nilpotent groups // Math. Zh. – Almaty, 2007. – №2 (24). – P. 95—102.
[11] Khisamiev N. G. Torsion-free constructive nilpotent Rp groups // Siberian Math. J. – 2009. – №1 (50). – P. 222—230.
[12] Khisamiev N. G. Hierarchies of torsion-free Abelian groups // Algebra and Logic. – 1986. – №2 (25). – P. 205—226.
[13] Khisamiev N. G. On positive and constructive groups // Siberian Math. J. – 2012. – №5 (53). – P. 1133—1146.
[14] Nurizinov М. K., Tyulyubergenev R. K., Khisamiev N. G. Computable torsion-free nilpotent groups of finite dimension
// Siberian Math. J. – 2014. – №3 (55). – P. 580—591.
[15] Kargopolov M. I., Merzlyakov Yu. I. Fundamentals of the Theory of Groups. — Moscow: Nauka, 1996.
[16] Mal’cev A. I. Algorithms and Recursive Functions. — Moscow: Nauka, 1986.
[17] Fuchs L. Infinite Abelian groups, Mir, trans. From English, Moscow, 1974.
Downloads
How to Cite
Nurizinov, M. K., Tyulyubergenev, R. K., & Khisamiev, N. G. (2015). On constructive nilpotent groups. Journal of Mathematics, Mechanics and Computer Science, 87(4), 35–46. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/290
Issue
Section
Mathematics
