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POLYCYCLIC GROUPS, ANALYTIC GROUPS AND ALGEBRAIC GROUPS

Published online by Cambridge University Press:  09 July 2002

MARCUS DU SAUTOY
MARCUS DU SAUTOY
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford OX1 3LB. [email protected]
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Abstract

Philip Hall proved that the group operation in a finitely generated nilpotent group can be described by polynomials in the coordinates with respect to a Mal'cev basis. In this paper we generalize this result to the class of polycyclic groups. Using the construction of a semi-simple splitting for a polycylic group as introduced by Dan Segal, we prove that the group operation is described by polynomial and exponential functions. Hall's result provided a powerful route to the understanding of the connection between the abstract nilpotent group and associated topological and algebraic groups. In the same manner, we apply our result for polycyclic groups to illuminate various constructions, due to Steve Donkin and Andy Magid, of topological and algebraic groups associated with a polycyclic group.

2000 Mathematical Subject Classification: 20F22, 20G15, 22E05.

Keywords

polycyclic groupalgebriac groupanalytic groupLie algebra
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 85 , Issue 1 , March 2002 , pp. 62 - 92
Copyright
2002 London Mathematical Society

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