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Growth of generating sets for direct powers of classical algebraic structures

Research output: Contribution to journalArticle

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.
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Original languageEnglish
Pages (from-to)105-126
Number of pages22
JournalJournal of the Australian Mathematical Society
Volume89
Issue number1
DOIs
Publication statusPublished - Aug 2010

    Research areas

  • Generating sets, Growth, Direct products, Algebraic structures, Universal algebra

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