On translation planes of orderq 3 that admit a collineation group of orderq 3 |
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Authors: | Kenzi Akiyama Chihiro Suetake |
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Affiliation: | (1) Department of Applied Mathematics, Fukuoka University, 814-01 Fukuoka, Japan;(2) Amagasaki-minami High School, 660 Amagasaki, Hyogo, Japan |
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Abstract: | Let II be a translation plane of orderq3, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq3 in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq2, whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp3, forp a prime, admitting a collineation groupG of orderp3 in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp3 or (ii) the Sherk plane of order 27. |
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Keywords: | 51A40 |
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