Linear-Length Mixed Identities for Finite Projective Unitary Groups
Manuscript 20 October 2026 · Online 20 October 2026
Abstract
Let \(q=p^e\) be a prime power and \(n\geq2\). We construct a nontrivial singular mixed identity for \(\operatorname{PSU}_n(q)\) of variable length at most \(16q+8p+12\). The bound is uniform in the dimension and includes characteristic two without an exception. A product of two unitary transvections has order dividing one of \(q-1\), \(q+1\) or \(2p\). A nested commutator combines these alternatives, while separate free conjugators ensure formal nontriviality. Together with the lower bound of Bradford, Schneider and Thom, this gives \(\Theta(q)\) for the length of a shortest mixed identity in the projective special unitary family. This is a complete theorem for the stated unitary-family scope, not a solution of all remaining Lie-type families or shortest nonsingular identities in the associated Oberwolfach questions. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority claim.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete unitary-family theorem
- Categories
- math.GR
- Manuscript
- 20 October 2026
- Online release
- 20 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Linear-Length Mixed Identities for Finite Projective Unitary Groups,” EulerSolve Research Papers, OWR-14299097-014, 2026. https://doi.org/10.5281/zenodo.23507216.
BibTeX
@misc{ferudun2026unitaryidentities,
author = {Ferudun, Alper},
title = {Linear-Length Mixed Identities for Finite Projective Unitary Groups},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-14299097-014/},
doi = {10.5281/zenodo.23507216},
note = {OWR-14299097-014; unrefereed preprint}
}