Normalizer of Twisted Chevalley Groups over Commutative Rings
Shripad M. Garge, Deep H. Makadiya
TL;DR
The paper addresses the problem of identifying normalizers of twisted Chevalley groups $G_{oldsymbol{ ext{π,σ}}}( initeΦ,R)$ and their elementary twisted subgroups $E'_{oldsymbol{ ext{π,σ}}}( initeΦ,R)$ inside the ambient group $G_{oldsymbol{ ext{π,σ}}}( initeΦ,S)$ for ring extensions $S$ of $R$. It develops a Lie-algebraic framework via the tangent algebra $T(E'_{oldsymbol{ ext{σ}}}(R))$ and proves a key lemma showing $E'_{oldsymbol{ ext{σ}}}( initeΦ,R)$ generates $M_n(R)$, enabling a transfer between group and matrix realizations. The main result asserts that, under specific Noetherian and unit-invertibility conditions on $R$, the normalizers of $G_{oldsymbol{ ext{π,σ}}}( initeΦ,R)$ and $E'_{oldsymbol{ ext{π,σ}}}( initeΦ,R)$ in $G_{oldsymbol{ ext{π,σ}}}( initeΦ,S)$ coincide; moreover, in adjoint type this common normalizer equals $G_{ ext{ad},oldsymbol{ ext{σ}}}( initeΦ,R)$. These findings lay groundwork for understanding automorphisms of twisted Chevalley groups over rings. The work blends twisted root-system machinery, tangent-algebra techniques, and characteristic-subgroup arguments to bridge algebraic and group-theoretic structures.
Abstract
Let $R$ be a commutative ring with unity. Consider the twisted Chevalley group $G_{π, σ} (Φ, R)$ of type $φ$ over $R$ and its elementary subgroup $E'_{π, σ} (Φ, R)$. This paper investigates the normalizers of $E'_{π, σ}(Φ, R)$ and $G_{π, σ}(Φ, R)$ in the larger group $G_{π, σ}(Φ, S)$, where $S$ is an extension ring of $R$. We establish that under certain conditions on $R$ these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to $G_{π, σ}(Φ, R)$.