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The nonexistence of projective planes of order 12 with a collineation group of order 9. (English) Zbl 1437.51004

The existence of projective planes of order not a prime power is a challenging problem. After many years, the smallest open case (order \(10\)) was settled by C. W. H. Lam et al. [Can. J. Math. 41, No. 6, 1117–1123 (1989; Zbl 0691.51003)].
The study of projective planes of order \(12\) was begun by Z. Janko and Tran van Trung [J. Comb. Theory, Ser. A 29, 254–256 (1980; Zbl 0446.51006)] and they proved in a series of articles structural properties for the collineation group of a putative projective planes of order \(12\).
This paper provides new constraints for the collineation group \(G\) of such a putative plane. The main result is the following.
Theorem. There are no projective planes of order \(12\) admitting a collineation group of order \(9\).
As a corollary, thanks to previous characterization results, we know now that, if \(G\) is a collineation group of a projective plane \(\pi\) of order 12, then \(G\) is cyclic and \(|G|\) divides \(3\) or \(4\).
The proofs rely on computer searches and on the analysis of the possible actions of \(G\) on the symmetric transversal design contained in \(\pi\).

MSC:

51E15 Finite affine and projective planes (geometric aspects)

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References:

[1] A. Aguglia and A. Bonisoli, On the non-existence of a projective plane of order 15 with an A4-invariant oval, Discrete Math. 288 (2004), 1-7. · Zbl 1087.51006
[2] R. P. Anstee, M. Hall Jr. and J. G. Thompson, Planes of order 10 do not have a collineation of order 5, J. Combin. Theory Ser. A 29 (1980), 39-58. · Zbl 0442.05013
[3] K. Akiyama and C. Suetake, The nonexistence of projective planes of order 12 with a collineation group of order 8, J. Combin. Des. 16 (2008), 411-430. · Zbl 1151.51004
[4] K. Akiyama and C. Suetake, On projective planes of order 12 with a collineation group of order 9, Australas. J. Combin. 43 (2009), 133-162. · Zbl 1155.51301
[5] T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vols. I and II, Cambridge University Press, Cambridge, 1999. · Zbl 0945.05005
[6] K. Horvatic-Baldasar, E. Kramer and I. Matulic-Bedenic, Projective planes of order 12 do not have an abelian group of order 6 as a collineation group, Punime Mat. 1 (1986), 75-81. · Zbl 0629.51016
[7] K. Horvatic-Baldasar, E. Kramer and I. Matulic-Bedenic, On the full collineation group of projective planes of order 12, Punime Mat. 2 (1987), 9-11. · Zbl 0656.51008
[8] R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. 1 (1949), 88-93. · Zbl 0037.37502
[9] R. C. Bose and S. S. Shrikhande, On the falsity of Euler’s conjecture about the non-existence of two orthogonal latin squares of order 4t + 2, Proc. Nat. Acad. Sci. 45 (1959), 734-737. · Zbl 0085.00902
[10] R. C. Bose, S. S. Shrikhande and E. T. Parker, Further results on the construction of mutually orthogonal latin squares and the falsity of Euler’s conjecture, Canad. J. Math. 12 (1960), 189-203. · Zbl 0093.31905
[11] D. Casiello, L. Indaco and G. P. Nagy, Sull’approccio computazionale al problema dell’esistenza di un piano proiettivo di ordine 10, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 57 (2010), 69-88. · Zbl 1227.51006
[12] GAP—Groups, Algorithms, Programming—System for Computational Discrete Algebra, GAP 4.10.0, https://www gap-system.org.
[13] C. Y. Ho, Projective planes of order 15 and other odd composite orders, Geom. Dedicata 27 (1988), 49-64. · Zbl 0646.51014
[14] D. R. Hughes and F. C. Piper, Projective Planes, Springer-Verlag, Berlin, Heidelberg, New Vork, 1973. · Zbl 0267.50018
[15] Z. Janko and T. van Trung, On projective planes of order 12 which have a subplane of order three I, J. Combin. Theory Ser. A 29 (1980), 254-256. · Zbl 0446.51006
[16] Z. Janko and T. van Trung, Projective planes of order 12 do not have a nonabelian group of order 6 as a collineation group, J. Reine Angew. Math. 326 (1981), 152-157. · Zbl 0452.51008
[17] Z. Janko and T. van Trung, Projective planes of order 10 do not have a collineation of order 3, J. Reine Angew. Math. 325 (1981), 189-209. · Zbl 0462.51010
[18] Z. Janko and T. van Trung, Projective planes of order 12 do not possess an elation of order 3, Stud. Sci. Math. 16 (1981), 115-118. · Zbl 0488.05018
[19] Z. Janko and T. van Trung, On projective planes of order 12 with an automorphism of order 13, Part I: Kirkman designs of order 27, Geom. Dedicata 11 (1981), 257-284. · Zbl 0467.51010
[20] Z. Janko and T. van Trung, On projective planes of order 12 with an automorphism of order 13, Part II: Orbit matrices and conclusion, Geom. Dedicata 12 (1982), 87-99. · Zbl 0479.51008
[21] Z. Janko and T. van Trung, The full collineation group of any projective plane of order 12 in a{2, 3} group, Geom. Dedicata 12 (1982), 101-110. · Zbl 0474.51007
[22] Z. Janko and T. van Trung, A generalization of a result of L. Baumert and M. Hall about projective planes of order 12, J. Combin. Theory Ser. A 32 (1982), 378-385. · Zbl 0485.05018
[23] Z. Janko and T. van Trung, Projective planes of order 12 do not have a four group as a collineation group, J. Combin. Theory Ser. A 32 (1982), 401-404. · Zbl 0492.51014
[24] M. J. Kallaher, Affine Planes with Transitive Collineation Groups, NorthHolland, New York, Amsterdam, Oxford, 1982. · Zbl 0485.51006
[25] S. Kang and J.-H. Lee, An explicit formula and its fast algorithm for a class of symmetric balanced incomplete block designs, J. Appl. Math. Comput. 19 (2005), 105-125. · Zbl 1079.05009
[26] C. W. H. Lam, The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), 305-318. · Zbl 0744.51011
[27] E. T. Parker, Construction of some sets of mutually orthogonal Latin squares, Proc. Amer. Math. Soc. 10 (1959), 946-949. · Zbl 0093.02002
[28] E. T. Parker, Orthogonal Latin squares, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 859-862. · Zbl 0086.02201
[29] C. Suetake, On projective planes of order 15 admitting a collineation of order 7, Geom. Dedicata 81 (2000), 61-86. · Zbl 0960.51005
[30] C. Suetake, The nonexistence of projective planes of order 12 with a collineation group of order 16, J. Combin. Theory Ser. A 107 (2004), 21-48. · Zbl 1061.51004
[31] S. H. Whitesides, Collineations of projective planes of order 10, part I and II, J. Combin. Theory Ser. A 26 (1979), 249-277. · Zbl 0444.51007
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