The subnormal structure of classical-like groups over commutative rings

Abstract

Let $n$ be an integer greater than or equal to $3$ and $(R,Δ)$ a Hermitian form ring where $R$ is commutative. We prove that if $H$ is a subgroup of the odd-dimensional unitary group $U_{2n+1}(R,Δ)$ normalised by a relative elementary subgroup $EU_{2n+1}((R,Δ),(I,Ω))$, then there is an odd form ideal $(J,Σ)$ such that $EU_{2n+1}((R,Δ),(JI^{k},Ω_{\min}^{JI^k}\overset{\cdot}{+}Σ\circ I^{k}))\leq H \leq CU_{2n+1}((R,Δ),(J,Σ))$ where $k=12$ if $n=3$ respectively $k=10$ if $n\geq 4$. As a conseqence of this result we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.