Associativity of the Commutator Operation in Groups
The study of associativity of the commutator operation in groups goes back to the work of F. W. 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 Guzm´an 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 nonsolvable 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.AMS Classification 2000: Primary: 20D05 ; Secondary: 20F16, 20N02, 20F38.