Automorphically Equivalent Elements of Finite Abelian Groups

Given a finite abelian group $G$ and elements $x,y\in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi (x)=y$ if and only if $G\mathrm{/}⟨x⟩\cong G\mathrm{/}⟨y⟩$. 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 $G\mathrm{/}⟨x⟩$ 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.