The subnormal structure of classical-like groups over commutative rings

TL;DR

This paper characterizes subgroups of odd-dimensional unitary groups over commutative rings that are normalized by elementary subgroups, establishing a sandwich theorem describing their structure in terms of odd form ideals.

Contribution

It proves a subnormal structure theorem for subgroups of odd-dimensional unitary groups over commutative rings, extending the understanding of their subgroup lattice.

Findings

Subgroups normalized by elementary groups are sandwiched between elementary and congruence subgroups.

The structure is described via odd form ideals with explicit bounds depending on the dimension.

A new sandwich theorem for subnormal subgroups in this setting is established.

Abstract

Let $n$ be an integer greater than or equal to $3$ and $(R,\Delta)$ a Hermitian form ring where $R$ is commutative. We prove that if $H$ is a subgroup of the odd-dimensional unitary group $U_{2n+1}(R,\Delta)$ normalised by a relative elementary subgroup $EU_{2n+1}((R,\Delta),(I,\Omega))$, then there is an odd form ideal $(J,\Sigma)$ such that $EU_{2n+1}((R,\Delta),(JI^{k},\Omega_{\min}^{JI^k}\overset{\cdot}{+}\Sigma\circ I^{k}))\leq H \leq CU_{2n+1}((R,\Delta),(J,\Sigma))$ 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.