The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5
-
Bettina Eick
und
Ann-Kristin Engel
Abstract
We consider the isomorphism problem for the finitely generated torsion free nilpotent groups of Hirsch length at most five. We show how this problem translates to solving an explicitly given set of polynomial equations. Based on this, we introduce a canonical form for each isomorphism type of finitely generated torsion free nilpotent group of Hirsch length at most 5 and, using a variation of our methods, we give an explicit description of its automorphisms.
References
[1] G. Baumslag, F. B. Cannonito, D. J. S. Robinson and D. Segal, The algorithmic theory of polycyclic-by-finite groups, J. Algebra 142 (1991), 118–149. 10.1016/0021-8693(91)90221-SSuche in Google Scholar
[2] W. A. de Graaf and A. Pavan, Constructing arithmetic subgroups of unipotent groups, J. Algebra 322 (2009), no. 11, 3950–3970. 10.1016/j.jalgebra.2009.03.027Suche in Google Scholar
[3] B. Eick, Computing with infinite polycyclic groups, Groups and Computation III (DIMACS 1999), De Gruyter, Berlin (2001), 139–153. 10.1515/9783110872743.139Suche in Google Scholar
[4] B. Eick, Orbit-stabilizer problems and computing normalizers for polycyclic groups, J. Symbolic Comput. 34 (2002), 1–19. 10.1006/jsco.2002.0540Suche in Google Scholar
[5] B. Eick and A.-K. Engel, Gap code related to the solution of the isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5, www.icm.tu-bs.de/~beick/so.html. Suche in Google Scholar
[6] B. Eick and A.-K. Engel, Hall polynomials for the torsion free nilpotent groups of Hirsch length at most 5, preprint (2016), https://arxiv.org/abs/1605.01591. Suche in Google Scholar
[7] B. Eick and W. Nickel, Polycyclic – Computing with polycyclic groups, preprint (2005). Suche in Google Scholar
[8] P. Faccin, W. A. de Graaf and W. Plesken, Computing generators of the unit group of an integral abelian group ring, J. Algebra 373 (2013), 441–452. 10.1016/j.jalgebra.2012.09.031Suche in Google Scholar
[9] F. J. Grunewald and R. Scharlau, A note on finitely generated torsion-free nilpotent groups of class 2, J. Algebra 58 (1979), no. 1, 162–175. 10.1016/0021-8693(79)90197-2Suche in Google Scholar
[10] F. Grunewald and D. Segal, Some general algorithms. I: Arithmetic groups, Ann. of Math. (2) 112 (1980), 531–583. 10.2307/1971091Suche in Google Scholar
[11] F. Grunewald and D. Segal, Some general algorithms, II: Nilpotent groups, Ann. of Math. (2) 112 (1980), 585–617. 10.2307/1971092Suche in Google Scholar
[12] P. Hall, The Edmonton Notes on Nilpotent Groups, Queen Mary College, London, 1969. Suche in Google Scholar
[13] D. F. Holt, B. Eick and E. A. O’Brien, Handbook of Computational Group Theory, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2005. 10.1201/9781420035216Suche in Google Scholar
[14] E. I. Khukhro, Nilpotent Groups and Their Automorphisms, De Gruyter Exp. Math. 8, De Gruyter, Berlin, 1993. 10.1515/9783110846218Suche in Google Scholar
[15] M. Newman, Integral Matrices, Pure Appl. Math. 45, Academic Press, New York, 1972. Suche in Google Scholar
[16] D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts in Math. 80, Springer, New York, 1982. 10.1007/978-1-4684-0128-8Suche in Google Scholar
[17] The GAP Group, GAP – Groups, Algorithms and Programming, Version 4.4, available from http://www.gap-system.org, 2005. Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Pseudo-free families of finite computational elementary abelian p-groups
- Cryptography from the tropical Hessian pencil
- Public-key cryptosystem based on invariants of diagonalizable groups
- The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5
- Log-space conjugacy problem in the Grigorchuk group
- Knapsack problem for nilpotent groups
Artikel in diesem Heft
- Frontmatter
- Pseudo-free families of finite computational elementary abelian p-groups
- Cryptography from the tropical Hessian pencil
- Public-key cryptosystem based on invariants of diagonalizable groups
- The isomorphism problem for torsion free nilpotent groups of Hirsch length at most 5
- Log-space conjugacy problem in the Grigorchuk group
- Knapsack problem for nilpotent groups
