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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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".