A characterization of \(L_ n(K)\) as a permutation group (Q1332635)

Let \(G\) be a doubly-transitive group on the set \(\Omega\) and let \(G_ \alpha\) be the subgroup fixing \(\alpha \in \Omega\). Suppose \(G_ \alpha\) has a normal subgroup \(A^ \alpha\) that is not semiregular on \(\Omega \backslash \{\alpha\}\). Suppose, furthermore, that \(A^ \alpha\) is isomorphic to a finite dimensional vector space over a field \(K\), the quotient \(N_ G (A^ \alpha)/C_ G (A^ \alpha)\) acts as a linear group on \(A^ \alpha\), and \(\Gamma^ G\) is a set of \(TI\)-subgroups of \(G\) where \(\Gamma\) is the set of one-dimensional subspaces of \(A^ \alpha\). In this paper it is proved that one of the following cases holds: (i) \(\text{PSL} (n,L) \leq G \leq P \Gamma L (n,L)\) for some \(n \geq 3\) and a skew field \(L\) that contains \(K\) as a subfield of finite index. (ii) \(E^ c_ 6 \leq G \leq \Aut ({\mathbf P} ({\mathbf C}))\) where \({\mathbf C}\) is a Cayley division algebra, \({\mathbf P} ({\mathbf C})\) is the Cayley plane related to \({\mathbf C}\) and \(E^ c_ 6\) denotes the normal subgroup of \(\Aut ({\mathbf P} ({\mathbf C}))\) that is generated by all elations. If \(\Omega\) is finite the conclusion of such result implies the conclusion of a classical O'Nan's theorem [\textit{M. O'Nan}, Math. Z. 127, 301-314 (1972; Zbl 0258.20003)].