A characterization of some spreads of order \(q^ 3\) that admit GL(2,q) as a collineation group (Q1119909)

The authors consider \(\pi:=F_ p^{3m}\oplus F_ p^{3m}\) where \(F_ p^{3m}\) is the vector space 3m-truples over GF(p), \(m=1,2,3,...\), and a field \({\mathcal F}\) of 3m\(\times 3m\)-matrices with entries from \(GF(q)\cup \{O_{3m}\}\) such that \({\mathcal F}\cong GF(q)\), and \(q=p^ m\). \(\pi\) becomes a GL(2,\({\mathcal F})\)-module by \[ (x\oplus y)\left[ \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right]:=(xA+yC)\oplus (xB+yC), \] with \(x,y\in F_ p^{3m}\), A,B,C,D\(\in {\mathcal F}.\) The main result is: Any spread \(\Gamma\) of \(\pi\) whose components are left invariant by the action of GL(2,\({\mathcal F})\) must be either Desarguesian or can be constructed from some fixed-point-free collineation \({\mathcal O}\in P\Gamma L(3,q)\setminus PGL(3,q)\) of the projective plane PG(2,q). In the latter case the method of construction yields in particular the spreads discovered by \textit{W. M. Kantor} [J. Comb. Theory, Ser. A 32, 299-302 (1982; Zbl 0485.51002), and Finite Geometries, Lect. Notes Pure Appl. Math. 82, 251-261 (1983; Zbl 0548.51011)] and \textit{C. Bartolone} and \textit{T. G. Ostrom} [J. Algebra 99, 50-57 (1986; Zbl 0591.51006)].