Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T09:59:05.361Z Has data issue: false hasContentIssue false
  • English
  • Français

Unification in Varieties of Groups:Nilpotent Varieties

Published online by Cambridge University Press:  20 November 2018

Michael H. Albert  and
John Lawrence
Michael H. Albert
Affiliation:
Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.
John Lawrence
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L3G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we show that any system of equations over a free nilpotent group of class c is either unitary or miliary. In fact, such a system either has a most general solution (akin to the most general solution of a system of linear dipohantine equations), or every solution has a proper generalization. In principle we provide an algorithm for determining whether or not a most general solution exists, and exhibiting it if it does.

Keywords

20F1820E1068Q40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1135 - 1149
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Burke, E. K., The undecidability of the unification problem for nilpotent groups of class > 5, J. London Math. Soc, to appear.+5,+J.+London+Math.+Soc,+to+appear.>Google Scholar
2. Chou, T. J. and Collins, G. E., Algorithms for the solution of systems of linear diophantine equations, SIAM J. Comput. 11(1982), 687–708.Google Scholar
3. Iliopoulos, Costas S., Worst-case complexity bounds on algorithms for computing the canonical structure of infinite abelian groups and solving systems of linear diophantine equations, SIAM J. Comput. 18(1989), 658–669.Google Scholar
4. Lawrence, J., The definability of the commutator subgroup in a variety generated by a finite group, Canad. Math. Bull. 28(1985), 505–507.Google Scholar
5. Lyndon, Roger C. and Schupp, Paul E., Combinatorial Group Theory, Springer Verlag, Berlin, Heidelberg, New York, 1977.Google Scholar
6. Neumann, H., Varieties of groups, Springer Verlag, Berlin, Heidelberg, New York, 1967.Google Scholar
7. Repin, N. N., Some simply presented groups for which an algorithm recognizing solvability of equations is impossible (Russian), Voprosy Kibernet. (Moscow) 134(1988), 167–175.Google Scholar
8. Roman‚kov, V. A., Unsolv ability of the endomorphic reducibility problem in free nilpotent groups and in free rings, Algebra and Logic 16(1977), 310–320.Google Scholar
9. Truss, J. K., Equation solving in free nilpotent groups of class 2 and 3, University of Leeds preprint series 24,1992.Google Scholar