On interpretability in varieties of groups (Q1070034)

A variety of groups W is said to be interpretable in a variety of groups V, written \(W\leq V\), if there is a word v(x,y) such that for every group G in V the set G with the derived operation * defined by \(x*y=v(x,y)\) is a group in W. The author makes several observations concerning pairs V and W satisfying \(V\leq W\) and \(W\leq V\). One result: if \(V\leq W\) and \(W\leq V\) then V and W have the same exponent. Another result: if V consists of all groups which are nilpotent of class 2 and have exponent n (odd) and W consists of all abelian groups of exponent n then \(V\leq W\) and \(W\leq V.\) \{Reviewer's note: some of the results in the paper were established by \textit{C. D. H. Cooper} [Proc. Second Internat. Conf. Theory of Groups, Canberra 1973, 221-225 (1974; Zbl 0294.20039)].\}