Fox derivations are an effective tool for studying free groups and their group rings. Let Fr be a free group of finite rank r with basis {f1,..., fr}. For every i, the partial Fox derivations ∂/∂fi и ∂/∂fi-1 are defined on the group ring ℤ[Fr]. For k / 2, their superpositions Dfϵi = = ∂/∂fϵki о ... о ∂/∂fϵk1, ϵ = (ϵ1,..., ϵk) Є{±1}k, are not Fox derivations. In this paper, we study the properties of superpositions Dfϵi. It is shown that the restrictions of such superpositions to the commutant F′r are Fox derivations. As an application of the obtained results, it is established that for any rational subset R of F′r and any i there are parameters k and ϵ such that R is annihilated by Dfϵi.
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- Title Superpositions of free Fox derivations
- Headline Superpositions of free Fox derivations
- Publesher
Tomsk State University
- Issue Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics 56
- Date:
- DOI 10.17223/20710410/56/3
Keywords
free group, group ring, Fox derivations, annihilators, rational subsetsAuthors
References
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Superpositions of free Fox derivations | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2022. № 56. DOI: 10.17223/20710410/56/3
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