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Varieties generated by wreath products of Abelian groups. (English. Russian original) Zbl 1283.20020
J. Math. Sci., New York 195, No. 4, 523-528 (2013); translation from Sovrem. Mat. Prilozh. 83 (2012).
Summary: This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups \(\mathfrak X\mathrm{\,Wr\,}\mathfrak Y=\{X\mathrm{\,Wr\,}Y\mid X\in\mathfrak X,\;Y\in\mathfrak Y\}\) and \(\mathfrak X\mathrm{\,wr\,}\mathfrak Y=\{X\mathrm{\,wr\,}Y\mid X\in\mathfrak X,\;Y\in\mathfrak Y\}\) generate the product variety \(\mathfrak X\mathrm{\,var}(\mathfrak Y)\).

MSC:
20E10 Quasivarieties and varieties of groups
20E22 Extensions, wreath products, and other compositions of groups
20K10 Torsion groups, primary groups and generalized primary groups
20F16 Solvable groups, supersolvable groups
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