Mechanical proofs of the Levi commutator problem
نویسنده
چکیده
This note presents purely mechanical proofs of the Levi commutator problem in group theory. The problem was solved first by using the theorem prover EQP, developed by William McCune at the Argonne National Laboratory. The fastest proof was found by using Peers-mcd, the Clause-Diffusion parallelization of EQP, developed by the author at the University of Iowa. 1 The Levi commutator problem The Levi commutator problem is an equational problem in group theory. Given the axioms for a group with product ∗ and identity e
منابع مشابه
A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids
This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...
متن کاملCommutator width in the first Grigorchuk group
Let G be the first Grigorchuk group. We show that the commutator width of G is 2: every element g ∈ [G, G] is a product of two commutators, and also of six conjugates of a. Furthermore, we show that every finitely generated subgroup H ≤ G has finite commutator width, which however can be arbitrarily large, and that G contains a subgroup of infinite commutator width. The proofs were assisted by ...
متن کاملLevi-Properties Generated by Varieties
Levi-properties were first introduced by L. C. Kappe and are modeled after groups investigated by F. W. Levi where conjugates commute. Let X be a group theoretic class. A group is in the derived class L(X) if the normal closure of each element in the group is an X -group. The property of being in the class L (X) is called the Levi-property generated by X . In the case where X is a variety, we s...
متن کاملPseudoconvextty and the Problem of Levi
The Levi problem is a very old problem in the theory of several complex variables and in its original form was solved long ago. However, over the years various extensions and generalizations of the Levi problem were proposed and investigated. Some of the more general forms of the Levi problem still remain unsolved. In the past few years there has been a lot of activity in this area. The purpose...
متن کاملAssociativity 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 Guzmán proved that a group G is solvable if and only if the commutator operation in G eventually satisfies ...
متن کامل