Automorphisms of triality groups over local rings
Abstract
Let R be a nonzero commutative local ring with 2,3∈ R×, and let ρ∈Aut(R) satisfy ρ3=id. We classify abstract automorphisms of the elementary and full triality groups in the adjoint and simply connected forms of type D4. Every automorphism is uniquely determined by an adjoint group element and a ring automorphism preserving the trace T=id+ρ+ρ2; conversely, every such pair defines an automorphism. A square-zero example shows that trace-preserving coefficient automorphisms need not normalize ⟨ρ⟩ when ρ is nontrivial on R but trivial on its residue field. We also prove a relative rigidity theorem for homomorphisms from elementary triality groups over arbitrary coefficient rings to local targets with $2$ and $3$ invertible, assuming that an adjoint conjugation makes the parameter-one root images standard modulo the maximal ideal. Conversely, every unital trace-preserving ring homomorphism into such a local coefficient ring induces the corresponding homomorphism of elementary groups.
Related
- Automorphisms of Twisted Chevalley Groups of type 2 A3 and 2 A4 over Local Rings
- Automorphisms of Twisted Chevalley Groups of type 2 Dℓ (ℓ ≥ 4) over Local Rings
- Automorphisms of twisted Chevalley groups of type 2E6 over local rings
- Automorphisms of Twisted Chevalley Groups over Commutative Rings
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BibTeX
@misc{bunina2026automorphisms,
title = {{Automorphisms of triality groups over local rings}},
author = {Elena Bunina and Avi Berezovskiy and Deep Makadiya},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21662},
eprint = {2610.21662},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21662},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Automorphisms of triality groups over local rings
AU - Elena Bunina
AU - Avi Berezovskiy
AU - Deep Makadiya
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21662
UR - https://arxiv.org/abs/2610.21662
AB - Let R be a nonzero commutative local ring with 2,3∈ R^×, and let ρ∈Aut(R) satisfy ρ^3=id. We classify abstract automorphisms of the elementary and full triality groups in the adjoint and simply connected forms of type D_4. Every automorphism is uniquely determined by an adjoint group element and a ring automorphism preserving the trace T=id+ρ+ρ^2; conversely, every such pair defines an automorphism. A square-zero example shows that trace-preserving coefficient automorphisms need not normalize ⟨ρ⟩ when ρ is nontrivial on R but trivial on its residue field. We also prove a relative rigidity theorem for homomorphisms from elementary triality groups over arbitrary coefficient rings to local targets with $2$ and $3$ invertible, assuming that an adjoint conjugation makes the parameter-one root images standard modulo the maximal ideal. Conversely, every unital trace-preserving ring homomorphism into such a local coefficient ring induces the corresponding homomorphism of elementary groups.
ER -