Automorphically Equivalent Elements of Finite Abelian Groups

2025/10/07 by Agarwal, Arjun, R.C. Chen, Chen, Rachel +3
Biochemistry, Genetics and Molecular Biology · Computer Science · #20K01 (Primary) #20K30 (Secondary) #Cellular Automata and Applications #DNA and Biological Computing #FOS: Mathematics #Group Theory (math.GR) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2510.06013

openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a finite abelian group G and elements x, y ∈ G, we prove that there exists ϕ∈ Aut(G) such that ϕ(x) = y if and only if G/⟨ x ⟩ ≅ G/⟨ y ⟩. This result leads to our development of the two fastest known algorithms to determine if two elements of a finite abelian group are automorphic images of one another. The second algorithm also computes G/⟨ x ⟩ in a near-linear time algorithm for groups, most feasible when the group has exponent at most 10²⁰. We conculde with an algorithm that computes the automorphic orbits of finite abelian groups.

Citations

• Automorphisms of finite Abelian groups

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• Automorphisms of finite Abelian groups
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