
An affine plane ~" is tangentially transitive with respect to a subplane ~re if ~r admits a collineation group which fixes ~'e pointwise and acts transitive on each of the tangent pencils to ~r o. Jha [6] has studied finite translation planes ~" which are tangentially transitive with respect to ~re and has shown that if the order of ~is not 16 then ]rr[ = Irrel 2 and 7r is a generalized Hall plane. If ]~r I = 16, 1r is a generalized Hall plane but the case that I~'~1 = 2 is possible. In [9], Johnson and Ostrom have shown that there is a unique tangentially transitive plane of order 16 and with respect to a subplane of order 2, Walker [14] has also independently obtained this same result. General ized Hall planes are precisely those planes of order q2 that may be derived from semifield planes whose coordinate semifields have middle nuclei which contain GF(q). Kleinfeld [10] has shown that there is exactly one semifield plane of order 16 with middle nucleus GF(4); the semifield plane with kern GF(4). In this article we determine the nonisomorphic planes which may be derived from the semifield plane of order 16 and kern GF(4). We show that there are exactly three such planes ~-,,, -n" E, 7rH: • "n is a generalized Hall plane which admits affine homologies and whose full collineation group has point orbits on the line at infinity of lengths 12, 3, 1, 1 or 12, 3, 2. ~r E is not a generalized Hall plane and admits affine elations with three centers whose full collineation group is isomorphic to $3 × P G L ( 3 , 2) and has infinite point orbits of lengths 3, 14. rr,, is a generalized Hall plane which admits an isomorphic collineation group as that of 7r E and whose action on £~ coincides with that of rrz.
Colliniation Group, 510.mathematics, Translation Planes, Derived Semifield Planes of Order 16, Finite affine and projective planes (geometric aspects), Combinatorial aspects of finite geometries, Affine Planes, Article
Colliniation Group, 510.mathematics, Translation Planes, Derived Semifield Planes of Order 16, Finite affine and projective planes (geometric aspects), Combinatorial aspects of finite geometries, Affine Planes, Article
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