The subnormal structure of classical-like groups over commutative rings

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 $E{U}_{2n+1}((R,\Delta ),(I,\Omega ))$, then there is an odd form ideal $(J,\Sigma )$ such that $E{U}_{2n+1}((R,\Delta ),(J{I}^k,{\Omega }_{\min }^{J{I}^k}\stackrel{\cdot }{+}\Sigma \circ I^k))\le H\le C{U}_{2n+1}((R,\Delta ),(J,\Sigma ))$ where $k=12$ if $n=3$ respectively $k=10$ if $n\ge 4$. As a conseqence of this result we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.

arXiv admin note: text overlap with arXiv:1801.00699