# Non-abelian variety of groups in which finite groups are abelian

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.

• 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.
– YCor
Dec 1 '19 at 10:15
• 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.
– YCor
Dec 1 '19 at 10:19