Minimal generating sets for some wreath products of groups

Abstract

Let d(G) denote the minimum of the cardinalities of the generating sets of the group G. Call a generating set of cardinality d(G) a minimal generating set for G. If A is a finitely generated nilpotent group, B a non-trivial finitely generated abelian group and A wr B is their (restricted, standard) wreath product, then it is proved (by explicitly constructing a minimal generating set for A wr B ) that d(AwrB) = max{l+d(A), d(A×B)} where A × B is their direct product.