A necessary condition for Kantor's theorem holding and its applications (Q2749018)

\textit{W. M. Kantor} [J. Algebra 62, 232-234 (1980; Zbl 0429.20004)] proved the following theorem: Let \(F\) be a finite field, and \(K\) an extension of \(F\) of degree \(r\). If \(X\) is an intermediate group between \(\text{GL}(1,K)\) and \(\text{GL}(r,F)\), then there exists an intermediate field \(E\) between \(F\) and \(K\) such that \(\text{GL}(d,E)\leq X\leq\text{GL}(d,E)\cdot\Aut K/E\), where \(d=\dim_EK\) and \(\Aut K/E\) is the Galois group of the extension \(K/E\). Does Kantor's theorem hold for an arbitrary field \(F\)? In this paper, the authors give a necessary condition for Kantor's theorem holding for an arbitrary field \(F\) which contains non-square elements. A counterexample and some applications are also given.