bibkey: aliabadi2026nonabeliantwist
authors: Mohsen Aliabadi
year: 2026
title: A nonabelian twist on differences of bijections
doi: 10.61091/um128-18
url: https://arxiv.org/html/2605.16478v2
claim: Problem 7.6 asks whether every finite nonabelian group admits a multiset of cardinality its order satisfying the abelianization obstruction but not quotient-realizable.
strata_touched:
  - D5/S3/FiniteGroups/AliabadiQuotientObstruction
license: citation-only
triage: anchor

A nonabelian twist on differences of bijections

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Mohsen Aliabadi, arXiv:2605.16478v2; journal DOI 10.61091/um128-18. The official journal article and the arXiv v2 version are the sources for Definition 2.1, Proposition 2.3, and the complete Problem 7.6 with its following paragraph.

Problem 7.6 reads: “Does every finite nonabelian group admit a multiset A of cardinality |G| satisfying the abelianization obstruction but not quotient-realizable?” Mathematical typography is transcribed here. The immediately following paragraph reads: “A positive answer would characterize finite abelian groups as precisely those finite groups for which the abelianization obstruction is the only obstruction.”

Quotient realizability means two bijections b,c from an index set of size |G| to G whose multiset of right quotients b(i)c(i)^(-1) equals A. Proposition 2.3 gives the necessary obstruction: the product, with multiplicities, after the canonical projection to G/[G,G] is one. Repeated identities are permitted. The question does not require a product-one ordering in G or an identity-free multiset.

The associated formal result uses one nonidentity element of [G,G] and |G|-1 identity entries. Its proof applies the canonical abelianization kernel and the impossibility that a permutation moves exactly one point. This is an application of those general results to the stated question, without a claim to a new general theorem or priority. The source is cited, not redistributed.