Overgroups in \(\text{GL}(nr,F)\) of \(\text{TU}(n,K,h)\) or \(\Omega(n,K,Q)\)
Let \(K\), \(F\) be division rings, with \(K\supset F\), and \(\dim_FK=r<\infty\) when we regard \(K\) as a left \(F\)-space. An \(n\)-dimensional left \(K\)-space \(V(n,K)\) can be regarded as an \(nr\)-dimensional left space \(V=V(nr,F)\) over \(F\), and thus \(\text{GL}(n,k)\) acting on \(V(n,K)\) is embedded in \(\text{GL}(nr,F)\) acting on \(V(nr,F)\). In this note we treat the overgroups in \(\text{GL}(nr,F)\) of a unitary group \(\text{TU}(n,K,f)\) or an orthogonal group \(\Omega(n,K,Q)\). We classify all these overgroups, provided the Witt index \(v(f)\geq 2\) or \(v(Q)\geq 2\). According to our results, each of these overgroups is precisely a unitary or an orthogonal group acting on the vector space structure \(V(nd,E)\) over an intermediate division ring \(E\) (\(F\subseteq E\subseteq K\), \(d=\dim_EK\)). In the special case where \(F\) is a maximal division subring of \(K\), we obtain certain classes of maximal subgroups of the unitary or orthogonal groups over \(F\).
scientific article; zbMATH DE number 558935