On varieties of groups generated by wreath products of Abelian groups (Q2738759)

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.NEWLINENEWLINENEWLINEIt 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).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00043].