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On wreath products of finitely generated Abelian groups. (English) Zbl 1059.20024
de Giovanni, F. (ed.) et al., Advances in group theory 2002. Proceedings of the intensive bimester dedicated to the memory of Reinhold Baer (1902–1979), Napoli, Italy, May–June 2002. Rome: Aracne (ISBN 88-7999-564-2/pbk). 13-24 (2003).
In an earlier paper [Abelian groups, rings and modules. Proceedings of the AGRAM 2000 conference, Perth, Australia, July 9-15, 2000. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 273, 223-238 (2001; Zbl 0981.20018)] the author established a criterion under which the wreath product of the Abelian groups \(A\) and \(B\) generates the product variety \(\text{var}(A)\text{var}(B)\). If one of the groups has infinite exponent this is always true, but in the case where both exponents are finite the condition is rather complicated.
The aim of this paper is to show that if \(A\) and \(B\) are finitely generated Abelian groups with finite exponents, then their wreath product generates the product variety if and only if the exponents are co-prime.
The paper ends with a section giving examples to show that one cannot hope to extend these results to non-Abelian groups.
For the entire collection see [Zbl 1031.20001].

20E22 Extensions, wreath products, and other compositions of groups
20E10 Quasivarieties and varieties of groups
20K01 Finite abelian groups
20K10 Torsion groups, primary groups and generalized primary groups
20K25 Direct sums, direct products, etc. for abelian groups