The study of associativity of the commutator operation in groups goes back to
some work of Levi in 1942. In the 1960's Richard J. Thompson created a group F
whose elements are representatives of the generalized associative law for an
arbitrary binary operation. In 2006, Geoghegan and Guzman proved that a group G
is solvable if and only if the commutator operation in G eventually satisfies
ALL instances of the associative law, and also showed that many non-solvable
groups do not satisfy any instance of the generalized associative law. We will
address the question: Is there a non-solvable group which satisfies SOME
instance of the generalized associative law? For finite groups, we prove that
the answer is no.Comment: 8 page
