Abstract
Let II be a translation plane of orderq 3 with kernel ⊒GF(q) that admits a collineation groupG of orderq 3 in the linear translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity.
In this paper, we show that any such translation plane II is one of the following types of planes:
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(i)
a semifield plane,
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(ii)
a desirable plane, or
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(iii)
an elusive plane of type I or type II.
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Akiyama, K.: On the Sherk plane, to appear inJ. Geom.
Akiyama, K. and Suetake, C.: On translation planes of orderq 3 that admit a collineation group of orderq 3,Geom. Dedicata 55 (1995), 1–57.
Biliotti, M., Jha, V., Johnson, N.L. and Menichetti, G.: A structure theory for two-dimensional translation planes of orderq 2 that admit collineation groups of orderq 2,Geom. Dedicata 29 (1989), 7–43.
Fink, J.B., Johnson, N.L. and Wilke, F.W.: A characterization of ‘likable’ translation planes,Rend. Circ. Mat. Palermo 32 (1983), 76–99.
Hughes, D.R. and Piper, F.C.:Projective Planes, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
Johnson, N.L. and Wilke, F.W.: Translation planes of orderq 2 that admit collineation group of orderq 2,Geom. Dedicata 15 (1984), 293–312.
Narayana Rao, M.L. and Satyanarayana, K.: On the Sherk's plane of order 27,Linear Algebra Appl. 74 (1986), 1–9.
Sherk, F.A.: A translation plane of order 27 with a small translation complement,Geom. Dedicata 9 (1980), 307–316.
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Akiyama, K. On translation planes of orderq 3 that admit a collineation group of orderq 3, II. Geom Dedicata 57, 171–193 (1995). https://doi.org/10.1007/BF01264936
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DOI: https://doi.org/10.1007/BF01264936