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On varieties of groups generated by wreath products of Abelian groups. (English) Zbl 0981.20018
Kelarev, A. V. (ed.) et al., 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).
A first result regarding the question when the variety generated by \(A\text{ wr }B\) is the product variety \(\text{var}(A)\text{var}(B)\) for \(A\) and \(B\) Abelian was obtained by C. Houghton: the answer is “yes” if \(A=C_m\), \(B=C_n\) and \(m\) and \(n\) are coprime. The author treats the general case and obtains (Theorem 6.1): the answer is “yes” if and only if either (i) at least one of the factors is not of finite exponent or (ii) if \(p\) is a common prime divisor of both exponents, the highest Ulm \(p\)-factor \(B[p^k]/B[p^{k-1}]\) of \(B\) is of infinite rank.
It is shown by examples that generalizations to other classes of groups (for instance to nilpotent groups of class 2) must be very restricted if they are at all possible (Example 6.3, 6.4).
For the entire collection see [Zbl 0960.00043].

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
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