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GROWTH OF GENERATING SETS FOR DIRECT POWERS OF CLASSICAL ALGEBRAIC STRUCTURES

Published online by Cambridge University Press:  21 September 2010

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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