### Associativity of the Commutator Operation in Groups

#### Abstract

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.#### Full Text:

PDFDOI: http://dx.doi.org/10.11606/issn.2316-9028.v3i2p231-240

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