Derived Semifield Planes Research Articles

Some new examples of derived semifield planes with affine elations

The main aim of this paper is to give some new examples of a class of semifields that has been introduced by Biliotti and Menichetti in [1]. The cases of the BM-semifields constructed starting fom the fields GF(8) and GF(16) are investigated with the aid of the computer.

Derived semifield planes of odd order admitting a dihedral group of affine homologies

Derived semifield planes of odd order admitting non-trivial involutorial affine homologies with more than one axis, are examined in detail, under the assumption that the group generated in the translation complement is dihedral. The whole structure of the semifields S coordinatizing such planes is determined. The class of the semifields S of dimension 4 over their centres is characterized.

Derived semifield planes with affine elations

Derived semifield planes admitting non trivial affine elations with more than one centre are examined in detail and several new examples of such plantes are constructed. A new characterization of the Hall planes of even order among derived semifield planes is also given.

On central collineations of derived semifield planes

LetG denote the collineation group generated by the set of all affine central collineations in a derived semifield plane. We present a characterization of the Hall planes in terms of the order ofG. This essentially allows the extension of the theorems of Kirkpatrick and Rahilly on generalized Hall planes to arbitrary derived semifield planes. That is, a derived semifield plane of order q2 is a Hall plane precisely when it admits q+1 involutory central collineations.

A note on the derived semifield planes of order 16

A note on the derived semifield planes of order 16

A note on the derived semifield planes of order 16

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.

On Elations of Derived Semifield Planes

On Elations of Derived Semifield Planes

Collineation groups of derived semifield planes III

Collineation groups of derived semifield planes III

Collineation groups of derived semifield planes III

Collineation groups of derived semifield planes III

Collineation groups of derived semifield planes II

Collineation groups of derived semifield planes II

Collineation groups of derived semifield planes

Collineation groups of derived semifield planes