Associativity of the Commutator Operation in Groups

TL;DR

This paper investigates the associativity properties of the commutator operation in groups, establishing that non-solvable finite groups do not satisfy any specific instance of the generalized associative law.

Contribution

It proves that no non-solvable finite group satisfies any particular instance of the generalized associative law for the commutator operation.

Findings

Non-solvable finite groups do not satisfy any instance of the generalized associative law.

The study connects associativity of commutators with group solvability.

Extends previous work on associativity in group theory.

Abstract

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.