Irreducible linear groups of degree four over a quaternion division algebra that contain a subgroup \(\text{diag}(T_3(K,\Phi_0),1)\).
Let \(D\) be a quaternion division algebra over a field \(F\) of characteristic \(\neq 2\). The author continues his series of investigations of particular classes of linear groups over \(D\) [see, for example, \textit{E. L. Bashkirov}, St. Petersbg. Math. J. 13, No. 3, 373-388 (2002); translation from Algebra Anal. 13, No. 3, 43-64 (2002; Zbl 1009.20059)]. Let \(K\) be a subfield of \(D\) and \(\Phi_0\) be a nondegenerate \(\sigma\)-skew-Hermitian form in \(3\) variables over \(K\) where \(\sigma\) is an involutory automorphism of \(K\). Denote the group generated by the transvections in the unitary group \(U_3(K,\Phi_0)\) by \(T_3(K,\Phi_0)\), and the corresponding block diagonal subgroup of \(\text{GL}_4(D)\) by \(\text{diag}(T_3(K,\Phi_0),1)\). The main result of the present paper is the following. Suppose that \(\Phi_0\) has Witt index \(1\), and that \(K\) contains a subfield \(K_0\) of \(F\) such that \(F\) is algebraic over \(K_0\) and each element of \(K_0\) is fixed by \(\sigma\). Then each irreducible subgroup \(X\) of \(\text{GL}_4(D)\) which contains \(\text{diag}(T_3(K,\Phi_0),1)\) has a normal subgroup belonging to one of five specified classes. The classes are too complicated to describe here.
scientific article; zbMATH DE number 2182059
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