Denominators and odd values of the automorphism ratio of a finite abelian group
Abstract
For a finite group G put α(G)=|Aut(G)|/|G|. Problem 21.97 of the Kourovka Notebook asks whether every positive rational is of this form; the answer is yes, by a recent theorem of Sureaux, which also settles the nilpotent case. For abelian groups the answer is no, and the exact image is an open project of McCulloch, who proved that the reduced denominator of α(A) is squarefree and that no odd prime occurs. We settle two pieces of that project. First, the set of reduced denominators of α(A), as A runs over all finite abelian groups, is exactly the set of cyclic numbers {b:gcd(b,φ(b))=1} — the orders at which every group is cyclic, OEIS A003277. This is strictly sharper than squarefreeness: 6 and 21 are squarefree and are not denominators. Second, we determine the odd part of the image exactly: an element of the abelian image has odd numerator and odd denominator if and only if it is 1, or 21, or (q-1)/(2q) or 3(q-1)/(2q) for a prime q ≡ 3 (mod 4). In particular the only odd integers in the abelian image are 1 and 21, the latter realized by (ℤ/2)³ with |Aut|=|GL(3,2)|=168; this contains McCulloch's theorem that no odd prime occurs, and is strictly stronger. Both infinite families do occur, realized by ℤ/2q and by (ℤ/2)² × ℤ/q for q ≡ 3 (mod 4). The list has a gap in it — no odd-over-odd abelian value lies strictly between 1/2 and 1 — and the gap excludes 3/5 and 5/7 from the abelian image outright, although 5 and 7 are cyclic numbers and both do occur as denominators. Both theorems are short arguments built on a proposition of McCulloch, and we say so plainly; what we claim is that the statements are new and that they answer a clean part of his question. Every statement we prove is machine-checked in Lean 4; the classical formula for |Aut(A)| that they rest on is taken as given, and every step resting on a finite exhaustive search is named together with the size of that search.
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Source snapshot 2026-09-07 03:53 UTC
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Claim ledger
Stated results
KA1known data2026-09-03
Compute-first gate, in the kernel: the closed form p^(Ev L) * Nv p L agrees with the literal Hillar-Rhea product of Proposition 2.1 of arXiv:2603.29299v2 for every partition of every n <= 6 and every p in 2,3,5,7 (120 cases, by decide)
KA2known2026-09-03
Ev L >= -1 for every well-formed type list, and Ev L <= 0 exactly for the trivial group, the cyclic p-groups, Z/p x Z/p, Z/p x Z/pᶠ (f >= 2) and (Z/p)³ – the exponent form of the source's Proposition 2.3
KA3routine2026-09-03
The reduced denominator of |Aut(A)|/|A| is the product of the primes p with Ev(type p) = -1 whose p divides no other block's prime-to-p part (AbType.rat_den, via Rat.den_divₑqₒf_coprime)
KA4known2026-09-03
McCulloch's Theorem 1.1 in the model: the reduced denominator of |Aut(A)|/|A| is squarefree
KA5candidate2026-09-03
NEW: the reduced denominator b of |Aut(A)|/|A| always satisfies gcd(b, phi(b)) = 1 – b is a cyclic number (OEIS A003277), i.e. an order at which every group is cyclic. Strictly sharpens McCulloch's Theorem 1.1: 6 and 21 are squarefree but are not denominators
KA6routine2026-09-03
Converse of KA5: every b with gcd(b, phi(b)) = 1 really is a reduced denominator, realized by the cyclic group Z/b, whose ratio phi(b)/b is already in lowest terms
KA7routine2026-09-03
Counting automorphisms from generator images: for GenData G k (k generators plus a normal form), Fintype.card (MulAut G) equals the number of k-tuples of images whose induced map is an injective homomorphism – a |G|ᵏ search in place of Mathlib's |G|! one
KA8candidate2026-09-03
NEW: the complete odd slice of the abelian image. If |Aut(A)|/|A| has odd numerator and odd denominator then it is 1, or 21, or (q-1)/(2q) or 3(q-1)/(2q) for a prime q = 3 mod 4; in particular the only odd INTEGERS in the image are 1 and 21, the latter realized by (Z/2)³ with |Aut| = |GL(3,2)| = 168
KA9known data2026-09-03
Bridge rows: Mathlib's own automorphism counts Fintype.card (MulAut G) = 4, 8, 168, 8 for C₈, C₂ x C₄, (Z/2)³, Z/15, and the four resulting ratios 1/2, 1, 21, 8/15 agree with the model's pratio
KA10known2026-09-03
Nonabelian controls: Fintype.card (MulAut (QuaternionGroup 2)) = 24, so |Aut(Q₈)|/|Q₈| = 3 is an odd prime; and Fintype.card (MulAut (DihedralGroup 4)) = 8, ratio 1
KA11known2026-09-03
McCulloch's Theorem 1.3 ('no odd prime can equal |Aut(G)|/|G| for a finite abelian G') derived as a corollary of KA8
KA12routine2026-09-03
Negative controls: 21 and 6 are squarefree but are provably NOT reduced denominators (so KA5 is not a restatement of squarefreeness); 3 and 15 ARE denominators, realized by Z/6 and Z/15; and the (q-1)/(2q) branch of KA8 is attained, at q = 3 by Z/6 with ratio 1/3
KA13measurement2026-09-03
MEASUREMENT: exhaustive GAP scan of alpha(G) = |Aut(G)|/|G| over all 10,177 groups of every order <= 319 except order 256. Every reduced denominator is a cyclic number (0 exceptions), so neither McCulloch's squarefree obstruction nor its sharpening KA5 is detectably abelian-specific at these orders, although Sureaux's positive answer to 21.97 forces both to fail at some larger order. The odd integers realized at order <= 255 are 1,3,5,7,9,11,13,15,21,23,27,29,33,35,39,41,51,53,55,63 against 1,21 for abelian groups; the smallest witness for an odd prime is Q₈ = SmallGroup(8,4) with alpha = 3
This ledger entry is reported in prose and is not bound to a Lean theorem.KA14routine2026-09-03
Realization of the two infinite odd families: alpha(Z/2q) = (q-1)/(2q) and alpha((Z/2)² x Z/q) = 3(q-1)/(2q), and both are quotients of two odd integers, for every prime q = 3 mod 4
KA15candidate2026-09-03
NEW: the odd slice of the abelian image, exactly. A rational r is |Aut(A)|/|A| for a finite abelian A AND has odd reduced numerator and odd reduced denominator if and only if r = 1, r = 21, or r = (q-1)/(2q) or 3(q-1)/(2q) for a prime q = 3 mod 4. This completes row KA8's forward implication to a biconditional
KA16routine2026-09-03
Controls and a gap corollary: no odd-over-odd automorphism ratio of a finite abelian group lies strictly between 1/2 and 1; in particular 3/5 and 5/7 are not |Aut(A)|/|A| for any finite abelian A, although 5 and 7 are cyclic numbers realized by Z/5 and Z/7; and alpha(Z/10) = 2/5 is not odd-over-odd, so the congruence q = 3 mod 4 in KA15 is not decoration
KA17measurement2026-09-03
MEASUREMENT correcting KA13's census size: the exhaustive GAP scan of alpha(G) over every order <= 319 except 256 covers 8,570 groups – 7,012 of order <= 255 and 1,558 of orders 257..319 – not 10,177. On a fresh re-run: 0 non-cyclic-number denominators; odd integers realized 1,3,5,7,9,11,13,15,21,23,27,29,33,35,39,41,51,53,55,63 for orders <= 255 and 1,3,7,9,11,15,33,35,39,65,69,75 for orders 257..319; and 0 groups of any kind with an odd-over-odd alpha strictly between 1/2 and 1
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Write α(G) = |Aut(G)|/|G|. Problem 21.97 of the *Kourovka Notebook* (No. 21, Novosibirsk 2026 = arXiv:1401.0300v46, source file 21tkt.tex line 14115), proposed by M. Tărnăuceanu:
- Snapshot
- 2026-09-07 03:53 UTC
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