Imprimitive maximal subgroups of finite orthogonal groups
Let \(V\) be a vector space of dimension \(n\) (where \(n\geq 4\) and even) over a finite field \(\mathrm{GF}(q)\). Suppose \(V\) is endowed with a quadratic form \(Q\) and \(V\) is an orthogonal direct sum of regular 2-dimensional subspaces \(V_i\), i.e. \(V=V_1\oplus\cdots\oplus V_s\) (where \(s=n/2)\). The Witt index of \((V,Q)\) is then \(s\) or \(s-1\). Let \(G_0\), \(G_1\), and \(G\) be the global stabilizers of \(\{V_1,V_2,\ldots,V_s\}\) in \(\Omega_n(q)\), \(\mathrm{SO}_n(q)\), and \(O_n(q)\), respectively. Here \(O_n(q)\), \(\mathrm{SO}_n(q)\), and \(\Omega_n(q)\) are the orthogonal group, the special orthogonal group, and the commutator subgroup, respectively. The author shows: 1) If \(n=4\), then \(G\) is maximal in \(O_n(q)\) except when \(q=2\) or \(3\), but \(G_0\) and \(G_1\) are not maximal in \(\Omega_n(q)\) and \(\mathrm{SO}_n(q)\), respectively. 2) If \(n\geq 6\), then \(G_0\), \(G_1\) and \(G\) are maximal in \(\Omega_n(q)\), \(\mathrm{SO}_n(q)\), and \(O_n(q)\), respectively, except when \(q=3\). 3) If \(n=4\), then \(\mathrm{PG}\) is maximal in \(\mathrm{PO}_n(q)\) except when \(q=2\) or \(3\), but \(\mathrm{PG}_0\) and \(\mathrm{PG}_1\) are not maximal in \(\mathrm{P}\Omega_n(q)\) and \(\mathrm{PSO}_n(q)\), respectively. 4) If \(n\geq 6\), then \(\mathrm{PG}_0\), \(\mathrm{PG}_1\), and \(\mathrm{PG}\) are maximal in \(\mathrm{P}\Omega_n(q)\), \(\mathrm{PSO}_n(q)\), and \(\mathrm{PO}_n(q)\), respectively, except when \(q=3\).
scientific article; zbMATH DE number 3977275