Quotient-realizable multisets in a finite group: the classification in S₃, counterexamples in D₈ and Q₈, and the abelianization obstruction in every finite nonabelian group
Abstract
A multiset A of |G| elements of a finite group G is quotient-realizable if A = {b(i)c(i)⁻¹} for two enumerations b, c of G; equivalently, if A = {φ(x)x⁻¹: x ∈ G} for some permutation φ of G. Hall proved in 1952 that for abelian G the obvious necessary condition — that the elements of A have trivial product in G/[G,G] — is also sufficient, and Aliabadi recently showed that it is not sufficient for S₃ and asked a list of questions about the nonabelian case. We answer four of them outright — his Problems 7.1, 7.2, 7.3 and 7.6 — and, with Hall's theorem, a fifth, his Problem 7.5. First, the obstruction fails to be sufficient in every finite nonabelian group: one commutator together with |G|-1 copies of the identity is a witness, so the groups in which the abelianization obstruction is the only obstruction are exactly the finite abelian groups. Second, we classify the quotient-realizable multisets of cardinality 6 in S₃: there are 146 of them, and they are the two-sided translates of ten explicit representatives. Third, for D₈ and for Q₈ we exhibit multisets that pass not only the abelianization test but also the strictly stronger product-one-ordering test that Aliabadi's S₃ example was built to pass, and are still not quotient-realizable; each comes with a human proof as well as an exhaustive one. Finally we count: of the 6435 multisets of cardinality 8, exactly 1423 are quotient-realizable in D₈ and exactly 1415 in Q₈, so this count separates two groups that share their order, their abelianization and their character degrees. The numbered results below are machine-checked in Lean 4 against Mathlib apart from four exceptions; Section [sec:verif] names those, says which of the remaining statements rest on exhaustive computation and how large each search was, and says which entries of the census are measurements rather than theorems.
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2b4eedb37d2c87166f474ef163e70a435b6d43aadb79b62a45ac12821a71f404
Claim ledger
Stated results
QR1known2026-09-03
Source Definition 2.1 (two bijections) and Lemma 2.3 (one permutation) are equivalent
QR2known2026-09-03
The abelianization condition is necessary, and already follows from a product-one ordering
QR3routine2026-09-03
A multiset with one non-identity element and |G|-1 copies of 1 is never quotient-realizable
QR4candidate2026-09-03
The source's Problem 7.6, answered: every finite nonabelian group admits a multiset satisfying the abelianization condition that is not quotient-realizable
QR5routine2026-09-03
Quotient-realizability is invariant under two-sided translation, inversion and automorphisms of G
QR6candidate2026-09-03
The source's Problem 7.1, answered: the quotient-realizable multisets of cardinality 6 in S₃ are exactly the two-sided translates of ten explicit representatives
QR7candidate2026-09-03
Exactly 146 of the 462 multisets of cardinality 6 in S₃ are quotient-realizable
QR8known2026-09-03
Source Proposition 5.4 and 5.6 reproduced: s,s,t,t,t,t is not quotient-realizable in S₃ although its elements multiply to 1 in some order
QR9known2026-09-03
Source Lemma 6.1 reproduced: a quotient-realizable multiset of cardinality 6 supported in s,t has 0, 3 or 6 copies of s, and each occurs
QR10known2026-09-03
Hall-Paige control: the multiset listing each group element once is quotient-realizable in D₈ and Q₈ but not in S₃
QR11candidate2026-09-03
The source's Problem 7.2, answered: s, s, sr, sr, sr, sr, sr, sr in D₈ satisfies the abelianization condition, admits a product-one ordering, and is not quotient-realizable
QR12routine2026-09-03
Non-vacuity control for QR11: replacing sr by sr², whose product with s has order 2, makes the multiset quotient-realizable
QR13candidate2026-09-03
The D₈ analogue of the source's Lemma 6.1: a quotient-realizable multiset of cardinality 8 supported in s, sr has 0, 4 or 8 copies of s
QR14candidate2026-09-03
Exactly 1423 of the 6435 multisets of cardinality 8 in D₈ are quotient-realizable
QR15candidate2026-09-03
The source's Problem 7.3, answered: a, a, x, x, x, x, x, x in Q₈ satisfies the abelianization condition, admits a product-one ordering, and is not quotient-realizable
QR16routine2026-09-03
Non-vacuity control for QR15: replacing x by x³, which generates the same cyclic subgroup, makes the multiset quotient-realizable
QR17candidate2026-09-03
The Q₈ analogue of the source's Lemma 6.1: a quotient-realizable multiset of cardinality 8 supported in a, x has 0, 4 or 8 copies of a
QR18candidate2026-09-03
Exactly 1415 of the 6435 multisets of cardinality 8 in Q₈ are quotient-realizable – eight fewer than in D₈
QR19measurement2026-09-03
The three-column census: 462/236/191/146 for S₃, 6435/1635/1521/1423 for D₈, 6435/1635/1517/1415 for Q₈
This ledger entry is reported in prose and is not bound to a Lean theorem.QR20routine2026-09-03
In-Lean enumeration of all multisets of a fixed cardinality over a finite type, with a proved completeness lemma
QR21known2026-09-03
Quotient-realizable implies a product-one ordering, in every finite group
QR22routine2026-09-03
The abelianization condition as an explicit homomorphism condition, instantiated at S₃ (sign), D₈ and Q₈ (C₂ x C₂)
QR23candidate2026-09-03
The S₃ census kernel-bound: 462 multisets of cardinality 6, 236 satisfying the abelianization condition, 191 with a product-one ordering, 146 quotient-realizable, and the gaps 90 and 45
QR24candidate2026-09-03
The D₈ census kernel-bound: 6435 multisets of cardinality 8, 1635 satisfying the abelianization condition, 1521 with a product-one ordering, 1423 quotient-realizable, and the gaps 212 and 98
QR25candidate2026-09-03
The Q₈ census kernel-bound: 6435 multisets of cardinality 8, 1635 satisfying the abelianization condition, 1517 with a product-one ordering, 1415 quotient-realizable, and the gaps 220 and 102
QR26routine2026-09-03
The abelianization column depends only on Gᵃb, the fibre size and n, by an explicit multiset formula – so D₈ and Q₈ must agree at 1635
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
- Founded 2026-09-03 from arXiv:2605.16478v2 (Mohsen Aliabadi, *A nonabelian twist on differences of bijections*, math.GR + math.CO; v1 15 May 2026, v2 26 Jun 2026, to appear in *Utilitas Mathematica*). The live version on 2026-09-03 is v2; the local corpus copy agrees with it.
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- 2026-09-07 03:53 UTC
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