An answer regarding automorphisms of finite abelian groups

Abstract: In this note we provide a negative answer to the question: ``Is it true that for every positive rational number $r$ there exists a finite abelian group $G$ such that $|\mathrm{Aut}(G)|/|G| = r$?". We show that if $r = a/b$ is a rational number (with $a$ and $b$ coprime integers) so that $r = |\mathrm{Aut}(G)|/|G|$ for a finite abelian group $G$, then $b$ is squarefree. We also show that no odd prime can equal $ |\mathrm{Aut}(G)|/|G|$ for a finite abelian group $G$.

• The paper proves that if |Aut(G)|/|G| = a/b in lowest terms, then b must be squarefree, imposing strict arithmetic constraints.
• Detailed analysis shows every power of 2 occurs while no odd prime quotient can appear, reflecting inherent group structure.
• Explicit constructions and systematic exclusions resolve an open Kourovka Notebook problem, confirming that not all positive rationals are achievable.

Automorphisms of Finite Abelian Groups and the Structure of Their Ratios

Overview and Main Results

This work addresses the following question: Does every positive rational number $r$ arise as the ratio $|\mathrm{Aut}(G)|/|G|$ for some finite abelian group $G$? While previous results established the density of such ratios in $[0, \infty)$, their surjectivity onto all positive rationals remained unresolved. The paper provides a complete negative answer to this question by establishing stringent number-theoretic constraints on possible denominators and the non-realizability of odd primes as these ratios.

The principal results can be summarized as follows:

• If $r = a/b$ with $\gcd(a, b) = 1$ and $r = |\mathrm{Aut}(G)|/|G|$ for some finite abelian group $G$, the denominator $b$ must be squarefree.
• Every power of $2$ can be realized as $|\mathrm{Aut}(G)|/|G|$0 for an appropriate finite abelian group $|\mathrm{Aut}(G)|/|G|$1.
• No odd prime $|\mathrm{Aut}(G)|/|G|$2 can be attained as $|\mathrm{Aut}(G)|/|G|$3 for any finite abelian group $|\mathrm{Aut}(G)|/|G|$4.

Analysis of Automorphism Group Sizes

The derivation hinges on a detailed analysis of the group $|\mathrm{Aut}(G)|/|G|$5 for finite abelian groups $|\mathrm{Aut}(G)|/|G|$6. Using canonical group decompositions, attention is reduced to primary abelian $|\mathrm{Aut}(G)|/|G|$7-groups. The well-known formula for $|\mathrm{Aut}(G)|/|G|$8 in the case of $|\mathrm{Aut}(G)|/|G|$9 permits explicit calculation of $G$0 in terms of the partition parameters $G$1.

Furthermore, the closure properties of $G$2 under the external direct product for coprime order factors establishes that for $G$3,

$G$4

where each $G$5 is a $G$6-group for distinct primes $G$7. The structural constraints on each $G$8 drive the main theorems.

Restrictions On the Denominator

A critical outcome is that the denominators in reduced fractions for possible values of $G$9 must be squarefree. This follows from the explicit checks for all possible $[0, \infty)$0-group types, and an analysis of the behavior of $[0, \infty)$1 under products. Each denominator $[0, \infty)$2 is either $[0, \infty)$3 or a single prime, and under multiplication across coprime types, the denominators multiply without introducing repeated prime factors, hence $[0, \infty)$4 must be squarefree.

Realizability of Certain Values

Explicit constructions demonstrate that all positive integer powers of $[0, \infty)$5 are attainable as ratios $[0, \infty)$6. The paper provides concrete examples (e.g., $[0, \infty)$7 yields $[0, \infty)$8), confirming the absence of obstructions in the $[0, \infty)$9-primary case.

Non-Realizability of Odd Primes

A key result is the non-occurrence of any odd prime $r = a/b$0 as a value of $r = a/b$1. The proof analyzes all possible abelian group types (cyclic, elementary abelian, and certain noncyclic forms) and exploits divisibility conditions arising in the automorphism formula (notably the presence of factors such as $r = a/b$2 and $r = a/b$3 in all nontrivial cases). Careful combinatorial analysis on possible group decompositions rules out the possibility of achieving an odd prime.

Implications and Directions for Future Work

The findings provide a decisive answer to a longstanding question in group theory (notably problem #21.97 in the Kourovka notebook). The result that denominators must be squarefree significantly narrows the set of admissible positive rationals, thus characterizing the arithmetical limitations inherent to $r = a/b$4 in the abelian case.

These results impact ongoing efforts to classify automorphism group behavior in broader algebraic structures (e.g., nilpotent groups, as mentioned in proposed directions). The restrictions found here motivate further investigation into which rational values arise as $r = a/b$5 for more general finite groups or families thereof, and suggest that the group-theoretical generalization of the Gauss formula is subject to deeper combinatorial and arithmetic constraints than previously anticipated.

Conclusion

This work offers a precise classification of obtainable rational ratios $r = a/b$6 for finite abelian groups, establishing the necessity for squarefree denominators and the impossibility for odd prime values. These results settle a previously open classification problem, raise new questions for nilpotent and general finite groups, and refine the arithmetic understanding of automorphism group cardinalities relative to group size in finite abelian cases (2603.29299).