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- Title
Dependence over subgroups of free groups.
- Authors
Rosenmann, Amnon; Ventura, Enric
- Abstract
Given a finitely generated subgroup H of a free group F, we present an algorithm which computes g 1 , ... , g m ∈ F , such that the set of elements g ∈ F , for which there exists a non-trivial H-equation having g as a solution is precisely the disjoint union of the double cosets H ⊔ H g 1 H ⊔ ⋯ ⊔ H g m H. Moreover, we present an algorithm which, given a finitely generated subgroup H ≤ F and an element g ∈ F , computes a finite set of elements from H * 〈 x 〉 (of the minimum possible cardinality) generating, as a normal subgroup, the "ideal" I H (g) ⊴ H * 〈 x 〉 of all "polynomials" w (x) , such that w (g) = 1. The algorithms, as well as the proofs, are based on the graph-theoretic techniques introduced by Stallings and on the more classical combinatorial techniques of Nielsen transformations. The key notion here is that of dependence of an element g ∈ F on a subgroup H. We also study the corresponding notions of dependence sequence and dependence closure of a subgroup.
- Subjects
FREE groups; POLYNOMIALS
- Publication
International Journal of Algebra & Computation, 2024, Vol 34, Issue 4, p439
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196724500176