Automorphically Equivalent Elements of Finite Abelian Groups

Abstract

Given a finite abelian group $G$ and elements $x,y\in G$, we prove that there exists $\ddot{I}•\in \text{Aut}(G)$ such that $\ddot{I}•(x)=y$ if and only if $G/⟨x⟩\cong G/⟨y⟩$. 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 $10^{20}$. We conculde with an algorithm that computes the automorphic orbits of finite abelian groups.

12 pages