Abstract
Let \(\Pi \) be a plane of order \(q^{3}\), \(q>2\), admitting \(G\cong PGL(3,q)\) as a collineation group. By Dempwolff (Geometriae Dedicata 18:101–112, 1985) the plane \(\Pi \) contains a G-invariant subplane \(\pi _{0}\) isomorphic to PG(2, q) on which G acts 2-transitively. In this paper it is shown that, if the homologies of \(\pi _{0}\) contained in G extend to \(\Pi \) then \(\Pi \) is either the desarguesian or the Figueroa plane.
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References
Biliotti, M., Montinaro, A.: On the rigidity of the Figueroa replacement in \(PG(2, q^{3})\). Combinatorica 37, 375–395 (2017)
Bruck, R.H.: Circle geometry in higher dimensions. In: A Survey of Combinatorial Theory (Proceedings of International Symposium, Colorado State University, Fort Collins, Colorado, 1971), pp. 69–77. North-Holland, Amsterdam (1973)
Bruck, R.H.: Circle geometry in higher dimensions II. Geom. Dedicata 2, 133–188 (1973)
Dembowski, P.: Finite Geometries. Reprint of the 1968. Original Classics in Mathematics. Springer, Berlin (1977)
Dempwolff, U.: \(PSL(3, q)\) on projective planes of order \(q^{3}\). Geom. Dedicata 18, 101–112 (1985)
Figueroa, R.: A family of not \((V, l)\)-transitive projective planes of order \(q^{3}\), \(q\equiv 1(mod3)\) and \(q>2\). Math. Z. 181, 471–479 (1982)
Hering, C., Schaeffer, H.J.: On the new projective planes of R. Figueroa. In: Combinatorial Theory, Schloss Rauischholzhausen, 1982. Lecture Notes in Mathematics, vol. 969, pp. 187–190. Springer, Berlin-New York (1982)
Hughes, D.R., Piper, F.C.: Projective Planes. Springer, New York, Berlin (1973)
Lavrauw, M., Van de Voorde, G.: On linear sets on a projective line. Des. Codes Cryptogr. 56, 89–104 (2010)
Lunardon, G., Polverino, O.: Translation ovoids of orthogonal polar spaces. Forum Math. 16, 663–669 (2004)
Lüneburg, H.: Characterizations of the generalized Hughes planes. Can. J. Math. 28, 376–402 (1976)
Ostrom, T.G., Wagner, A.: On projective and affine planes with transitive collineation groups. Math. Z. 71, 186–199 (1959)
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The authors are grateful to the referees for their valuable comments and suggestions.
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Biliotti, M., Montinaro, A. A characterization of the desarguesian and the Figueroa planes of order \(q^{3}\) . J Algebr Comb 48, 549–563 (2018). https://doi.org/10.1007/s10801-017-0802-2
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DOI: https://doi.org/10.1007/s10801-017-0802-2