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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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