Linear groups over a skew field of quaternions containing the group \(T_n(A,\Phi)\) over a subsfield
The aim of this paper is to describe the subgroups of the general linear group over a skew field of quaternions containing a given subgroup. More concretely, let \(D\) be a quaternion sfield over a field \(F\) of characteristic \(\neq 2\), and \(A\) a subsfield of \(D\), i.e., \(A\) is a quaternion sfield over a subfield \(k\) of \(F\). Let \(n\geq 3\) be an integer, \(E\) a right \(n\)-dimensional \(D\)-vector space, \(J\) an involution of \(D\) such that the set of all \(J\)-symmetric elements in \(D\) coincides with \(F\), and \(\Phi\) a non-degenerate \(J\)-skew-Hermitean form on \(E\) of Witt index \(\geq 1\). Denote by \(U_n(D,\Phi,J)\) the unitary group of \(\Phi\) and by \(T_n(D,\Phi,J)\) the normal subgroup of \(U_n(D,\Phi,J)\) generated by all transvections in \(U_n(D,\Phi,J)\). Assume \(D\) is algebraic over \(k\). If \(X\) is a subgroup of \(\text{GL}_n(D)\) containing \(T_n(A,\Phi,J)\), then \(X\) contains a normal subgroup \(G\) such that \(G=\text{SL}_n(L)\) or \(G=T_n(L,\Phi_{(L)},J')\), where \(L\) is a quaternion subsfield with \(A\subseteq L\subseteq D\) and \(J'\) is an involution of \(D\).
scientific article; zbMATH DE number 1887546