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Is there a non-abelian variety of groups $V$ such that any finite group from $V$ is abelian?

This was posed in a paper by Hanna Neumann (1967), but I cannot find the solution.

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    $\begingroup$ OP didn't react to the discussion. I edited by removing the first question because (1) it has an obvious answer (e.g., the the free group on 2 generators) (2) even with additional hypotheses as suggested by several users (e.g., not torsion-free, or torsion) it has easy or standard examples (appearing at various places here at more focussed question) so a separate answer would not be very useful (3) asking 2 independent questions in the same post is usually not recommended (4) the second question is more interesting and has an accepted answer. $\endgroup$
    – YCor
    Commented Dec 1, 2019 at 10:15
  • $\begingroup$ Trying to track the OP's reference, there are 2 references by Hanna Neumann in 1967 called Varieties of groups. The first is a whole 192-page book published at Springer (doi.org/10.1007/978-3-642-88599-0). The second is published at Proc. Internat. Conf. Theory of Groups (Canberra, 1965) 251-259 Gordon and Breach, New York. $\endgroup$
    – YCor
    Commented Dec 1, 2019 at 10:19

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The answer to Neumann's question is yes. A variety was constructed by Olshanskii , TY - JOUR AU - Ol'shanskiĭ, A., Varieties in which all finite groups are abelian DO - 10.1070/SM1986v054n01ABEH002960 Mathematics of the USSR-Sbornik He also constructed nonabelian varieties where every periodic group is abelian. I think all these can be found in Olshanskii's book "Geometry of defining relations".

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