2023/09/15 by Ruiwen Dong, Dong, Ruiwen
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Group Theory (math.GR) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2309.08811
openalex publication_date 2023/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups ⟨ G ⟩, ⟨ H ⟩ of a group G, decide whether the intersection ⟨ G ⟩ ∩ ⟨ H ⟩ is trivial. The second problem is Coset Intersection: given two finitely generated subgroups ⟨ G ⟩, ⟨ H ⟩ of a group G, as well as elements g, h ∈ G, decide whether the intersection of the two cosets g ⟨ G ⟩ ∩ h ⟨ H ⟩ is empty. We show that both problems are decidable in finitely generated abelian-by-cyclic groups. In particular, we reduce them to the Shifted Monomial Membership problem (whether an ideal of the Laurent polynomial ring over integers contains any element of the form Xz - f, z ∈ ℤ ∖ \0\). We also point out some obstacles for generalizing these results from abelian-by-cyclic groups to arbitrary metabelian groups.