Automorphically Equivalent Elements of Finite Abelian Groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917159029964800 |
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| author | Agarwal, Arjun Chen, Rachel Garg, Rohan Kettinger, Jared |
| author_facet | Agarwal, Arjun Chen, Rachel Garg, Rohan Kettinger, Jared |
| contents | Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $ϕ\in \text{Aut}(G)$ such that $ϕ(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. 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 $G/\langle x \rangle$ in a near-linear time algorithm for groups, most feasible when the group has exponent at most $10^{20}$. We conculde with an algorithm that computes the automorphic orbits of finite abelian groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_06013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphically Equivalent Elements of Finite Abelian Groups Agarwal, Arjun Chen, Rachel Garg, Rohan Kettinger, Jared Group Theory 20K01 (Primary), 20K30 (Secondary) Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $ϕ\in \text{Aut}(G)$ such that $ϕ(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. 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 $G/\langle x \rangle$ in a near-linear time algorithm for groups, most feasible when the group has exponent at most $10^{20}$. We conculde with an algorithm that computes the automorphic orbits of finite abelian groups. |
| title | Automorphically Equivalent Elements of Finite Abelian Groups |
| topic | Group Theory 20K01 (Primary), 20K30 (Secondary) |
| url | https://arxiv.org/abs/2510.06013 |