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Polycyclic Groups, Analytic Groups and Algebraic Groups
Author(s) -
Sautoy Marcus Du
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/85.1.62
Subject(s) - mathematics , nilpotent , group (periodic table) , algebraic number , algebraic group , simple (philosophy) , pure mathematics , class (philosophy) , connection (principal bundle) , group theory , polynomial , algebra over a field , topology (electrical circuits) , combinatorics , geometry , computer science , mathematical analysis , philosophy , chemistry , organic chemistry , epistemology , artificial intelligence
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.