Growth of generating sets for direct powers of classical algebraic structures
Date
08/2010Keywords
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Abstract
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.
Citation
Quick , M & Ruskuc , N 2010 , ' Growth of generating sets for direct powers of classical algebraic structures ' , Journal of the Australian Mathematical Society , vol. 89 , no. 1 , pp. 105-126 . https://doi.org/10.1017/S1446788710001473
Publication
Journal of the Australian Mathematical Society
Status
Peer reviewed
ISSN
1446-7887Type
Journal article
Rights
(c) 2010 Australian Mathematical Publishing Association Inc.
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