GROWTH OF GENERATING SETS FOR DIRECT POWERS OF CLASSICAL ALGEBRAIC STRUCTURES

Abstract

AbstractFor an algebraic structure A denote byd(A) the smallest size of a generating set for A, and letd(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 sequenced(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite thend(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 thend(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.