Growth of generating sets for direct powers of classical algebraic structures

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


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


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