On Properties of Certain Geometric Structures

"Affine planes, projective planes, spreads, translation planes, semifield planes and Desarguesian planes were extensively
investigated by Buekenhout [1], Dembowski [2], Hering [3], Huhges and Piper [4], Kallaher [5], Knuth [6], Ostrom and
Wagner [7], Johnson [8] and recently by Al Ali Bani-Ata [9], [10], [11], [12], [13], [14], [15] and [16] Let Π = (P,L) be the
affine plane embedded into a projective plane Π ̂ = (P ̂,L) by adjoining a certain line L ̂∞. If G is the collineation group of
Π and G*is the collineation group of Π ̂, then the first purpose of this proposal is to find the relation between G* and G ̂L
̂ ∞ the stabilizer of L ∞̂ in G*.Second we will study the relations among Aut(Π ̂ ) the collineation group of Π ̂, the elations
E (Π ̂ ) having axis L ∞̂ and (Aut(Π ̂ )) L ∞̂ the stabilizer group of L ∞̂ in the collineation group of Π ̂. Thus we are lead to
Hering's question, that is if G* is a given group, can one find a vector space V = V (2n, Fq) and aspread K of V such
that G* ≤ GK where GK = { gЄGL(V)| gleavesKinvariant } ? Third we formulate Hering's question as follows: We could
try to find a semifield A with G* ≤ Aut(A) and investigate the relation between Aut(A) and GK, where K is the spread
corresponding to A. Furthermore we will investigate the relation between G and the autotopism group GL0, L∞of Π (K)
and the translation plane corresponding to K, and L0 , L∞ are two components in K defined by
{(0,x)|xЄA} and {(x,0)|xЄA} respectively."