Automorphically Equivalent Elements of Finite Abelian Groups

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.

💡 Research Summary

The paper addresses the problem of determining when two elements of a finite abelian group are related by an automorphism. The central theoretical contribution is Theorem 2.1, which proves that for a finite abelian group (G) and elements (x, y\in G) there exists an automorphism (\phi\in\operatorname{Aut}(G)) with (\phi(x)=y) if and only if the quotient groups (G/\langle x\rangle) and (G/\langle y\rangle) are isomorphic. The proof proceeds by first showing the “only‑if” direction: any automorphism induces a well‑defined isomorphism between the corresponding coset groups. The converse is more involved; the authors decompose (G) into its Sylow‑(p) subgroups, reducing the problem to the case of a finite abelian (p)‑group. In that setting they write the group as a direct sum of cyclic (p)-power components, normalize the elements to vectors of the form ((p^{f_1},\dots,p^{f_n})), and use the hypothesis that the quotients are isomorphic to deduce that the multisets of exponents ({f_i}) coincide. From this they construct two explicit automorphisms: (\phi_1) maps the cyclic subgroup generated by (x) onto that generated by (y) (up to a scalar factor (k)), and (\phi_2) rescales each coordinate by the modular inverse of (k) (which exists because (\gcd(k,p)=1)). The composition (\phi_2\circ\phi_1) sends (x) to (y), completing the proof.

The paper also derives immediate corollaries. Theorem 2.4 shows that any two elements of maximal order in (G) are automorphic, and Corollary 2.5 follows that their corresponding quotients are automatically isomorphic. These results give a clean, structural criterion for automorphic equivalence that is easy to check in practice.

From the algorithmic perspective, the authors propose two methods to test the quotient‑isomorphism condition efficiently.

1. Direct Smith Normal Form (SNF) method – Represent (G) as (\bigoplus_{i=1}^k C_{m_i}) and form the ((k+1)\times k) integer matrix (\begin{bmatrix}x_1&\dots&x_k\ m_1&\dots&m_k\end{bmatrix}). Computing its SNF yields the invariant factors of (G/\langle x\rangle). Repeating for (y) and comparing the invariant factor lists decides automorphic equivalence. Using fast matrix multiplication (e.g., Strassen) the time complexity is (O(k^{2.8074})). This approach is straightforward and dominates for groups of modest rank (≈ 400 or less).

2. Sylow‑(p) decomposition with a specialized linear‑time routine – First factor the exponent of (G) (the largest order of any element) to split (G) into its Sylow‑(p) components. For each (p)-group (H\cong\bigoplus_{i=1}^r C_{p^{e_i}}) and an element (x) expressed as ((p^{f_1},\dots,p^{f_r})) (with (f_i) sorted), the algorithm iteratively adjusts the exponent list by adding (\max(0,e_i-f_i)) to all later entries and discarding the larger of each pair, ultimately leaving a single exponent per column that describes the invariant factors of (H/\langle x\rangle). Sorting the exponent list costs (O(r\log r)); the subsequent “range‑add” updates can be performed in linear time because they always affect suffixes. Summed over all primes, the total cost is (O(n\log n\log a_n)), where (n) is the rank of (G) and (a_n) its maximal exponent. When the exponent is bounded by (10^{20}), the dominant cost is the integer factorisation of (a_n), which has sub‑exponential complexity (\exp\bigl((64/9)^{1/3}(\log a_n)^{1/3}(\log\log a_n)^{2/3}\bigr)). Empirical analysis shows that for groups with rank above roughly 400 (or for smaller exponents), this Sylow‑based method outperforms the direct SNF approach.

The authors also prove that (|\operatorname{Aut}(G)|) grows exponentially with the rank (n) (Proposition 3.3). By reducing to the (p)-group case and using known formulas for automorphism groups of cyclic (p)-power modules, they show (|\operatorname{Aut}(G)|\ge c^{,n}) for some constant (c>1). Consequently, a naïve brute‑force algorithm that enumerates all automorphisms and checks (\phi(x)=y) would have exponential time in (n), whereas the proposed quotient‑isomorphism tests run in polynomial (or near‑linear) time.

Finally, the paper combines the two quotient‑testing algorithms to compute the full set of automorphic orbits (the “orbit partition") of a finite abelian group. By processing each element, computing its quotient invariants, and hashing the resulting invariant tuple, the algorithm groups together all elements lying in the same orbit. The overall runtime is essentially the sum of the per‑element quotient computations, yielding a near‑linear algorithm in the size of the group representation.

In summary, the work provides a clean algebraic characterisation of automorphic equivalence in finite abelian groups, translates this characterisation into two concrete algorithms with rigorous complexity analyses, and demonstrates that these algorithms dramatically improve over existing brute‑force methods, especially for groups of moderate to large rank. The results have potential applications in computational group theory, cryptography, and any domain where the structure of abelian groups and their automorphisms play a role.