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GROWTH OF GENERATING SETS FOR DIRECT POWERS OF CLASSICAL ALGEBRAIC STRUCTURES

Part of: Algebraic structures Lie algebras and Lie superalgebras Special aspects of infinite or finite groups

Published online by Cambridge University Press:  21 September 2010

MARTYN QUICK  and
N. RUŠKUC
MARTYN QUICK
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (email: [email protected])
N. RUŠKUC*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.

Keywords

generating setsgrowthdirect productsalgebraic structuresuniversal algebra

MSC classification

Secondary: 08A40: Operations, polynomials, primal algebras 17B99: None of the above, but in this section 20F05: Generators, relations, and presentations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 1 , August 2010 , pp. 105 - 126
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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