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Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

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Abstract

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq 2 havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq n, whereq is an odd prime power andn ≥ 3 such thatq n ≢ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq n, whereq is an odd prime power andn ≥ 2 such thatq n ≡ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq n such thatn ≡ 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].

The author is grateful to the referee for many helpful comments.

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  1. Amagasakiminami High School, 660, Amagasaki, Hyogo, Japan

    Chihiro Suetake

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  1. Chihiro Suetake
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Suetake, C. Flag transitive planes of orderq n with a long cyclel ∞ as a collineation. Graphs and Combinatorics 7, 183–195 (1991). https://doi.org/10.1007/BF01788143

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  • DOI: https://doi.org/10.1007/BF01788143

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