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Metabelian varieties of groups and wreath products of Abelian groups. (English) Zbl 1124.20017
When is the variety of the product $$XY$$ of two group classes equal to the variety of the (direct or Cartesian) wreath products of the varieties of $$X$$ and $$Y$$? This is answered here for Abelian group classes $$X$$ and $$Y$$, generalizing results of Higman and Houghton about cyclic groups. The case treated incompletely so far was the case that the exponents of $$X$$ and $$Y$$ are finite and not co-prime. The needed condition here, a statement on $$p$$-ranks needed among groups belonging to $$Y$$ for common prime divisors $$p$$, is given in Theorem 8.1.

##### MSC:
 20E10 Quasivarieties and varieties of groups 20E22 Extensions, wreath products, and other compositions of groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20K25 Direct sums, direct products, etc. for abelian groups
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