A problem of A. I. Mal'tsev (Q1822625)

As was shown by \textit{Yu. G. Klejman} [in Tr. Mosk. Mat. O.-va 44, 62-108 (1982; Zbl 0495.20013)] there exists a finitely-based variety of groups \(M\leq {\mathfrak A}^ 7\), free noncyclic groups of which have unsolvable word problem. It follows that the problem of implication of identities is unsolvable for the variety \({\mathfrak A}^ 7\). The author presents a simpler proof of unsolvability of the last problem for the variety \({\mathfrak A}^ 5\). Some ideas by the reviewer are used in this proof. Also it is shown that the isomorphism problem is unsolvable in the class of all 2-generated groups defined in \({\mathfrak A}^ 5\) by finitely many identities. An example is given of a 2-generated finitely based in \({\mathfrak A}^ 5\) relatively free but non residually finite group. Earlier known such examples were defined by infinite sets of identities.