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Symbolic Collection using Deep Thought

Published online by Cambridge University Press:  01 February 2010

C. R. Leedham-Green  and
Leonard H. Soicher
C. R. Leedham-Green
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, U.K., [email protected]
Leonard H. Soicher
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, U.K., [email protected]

Abstract

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We describe the “Deep Thought” algorithm, which can, among other things, take a commutator presentation for a finitely generated torsion-free nilpotent group G, and produce explicit polynomials for the multiplication of elements of G. These polynomials were first shown to exist by Philip Hall, and allow for “symbolic collection” in finitely generated nilpotent groups. We discuss various practicalissues in calculations in such groups, including the construction of a hybrid collector, making use of both the polynomials and ordinary collection from the left.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 1 , 1998 , pp. 9 - 24
Copyright
Copyright © London Mathematical Society 1998

References

1. Dekimpe, K. and Igodt, P., ‘Computational aspects of affine representations for torsion free nilpotent groups via the Seifert construction’, J. Pure Appl. Algebra 84 (1993) 165–190.CrossRefGoogle Scholar
2. Hall, P., ‘A contribution to the theory of groups of prime-power order’, Proc. London Math. Soc. (2) 36 (1934) 29–95.CrossRefGoogle Scholar
3. Hall, P., ‘Nilpotent groups’, Notes of lectures given at the Canadian Mathematical Congress 1957 Summer Seminar, in The collected works of Philip Hall (Clarendon Press, Oxford, 1988) pp. 415–462.Google Scholar
4. Leedham-Green, C.R. and Soicher, L.H., ‘Collection from the left and other strategies’, J. Symbolic Comp. 9 (1990) 665–675.CrossRefGoogle Scholar
5. Merkwitz, W.W., ‘Symbolische multiplikation in nilpotenten Gruppen mit Deep Thought’, Diplomarbeit, RWTH Aachen, 1997.Google Scholar
6. Robinson, D.J.S., A course in the theory of groups (Second Edition) (Springer, New York and Berlin, 1996).CrossRefGoogle Scholar
7. Schönert, M. et al., ‘GAP: groups, algorithms and programming’, version 3, release 4, Lehrstuhl D für Mathematik, RWTH Aachen, 1994.Google Scholar
8. Sims, C.C., Computation with finitely presented groups (Cambridge University Press, Cambridge, 1994).CrossRefGoogle Scholar