Projective planes and dihedral groups
Let \(q\) be a prime power and \(P\) the projective plane of order \(q^2\), \(B_0\), \(B_1\) two disjoint Baer subplanes of \(P\), \(\tau_0\), \(\tau_1\) the corresponding Baer involutions, \(\delta =\tau_0 \tau_1\). The author's main result is the following (a) \(\delta\) is a fixed point free projective collineation whose order \(r\) divides \(q^2- q+1\). (b) \(\delta= \sigma^n\) for some Singer cycle \(\sigma\) and some natural number \(n\). (c) \({\mathcal P}= \{\delta (B_0)^j \mid j\in \{0, \dots, r-1\}\}\) is a set of mutually disjoint Baer subplanes. If furthermore \(r=q^2 -q+1\) then \({\mathcal P}\) contains \(B_0\) and \(B_1\) and it coincides with the linear partition induced by \(\sigma\), cf. Theorem 4.3.5 in \textit{J. W. P. Hirschfeld} [`Projective geometry over finite fields' (1979; Zbl 0418.51002)]. The point orbits under the action of \(\langle \delta \rangle\) are complete \((q^2- q+1)\)-arcs of a special form described by \textit{B. C. Kestenband} [Geom. Dedicata 11, 107-117 (1981; Zbl 0452.51007)]. (d) \(D= \langle \tau_0, \tau_1 \rangle\) is a dihedral group of order \(2r\) whose reflections are the Baer involutions arising from the Baer subplanes in \({\mathcal P}\).
scientific article; zbMATH DE number 1110040
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