TAILIEUCHUNG - Báo cáo toán học: "Subplanes of order 3 in Hughes Planes"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Subplanes of order 3 in Hughes Planes. | Subplanes of order 3 in Hughes Planes Cafer Caliskan Department of Mathematical Sciences Michigan Technological University Houghton MI 49931 . ccaliska@ G. Eric Moorhouse Department of Mathematics University of Wyoming Laramie WY 82071 . moorhous@ Submitted Nov 30 2009 Accepted Nov 23 2010 Published Jan 5 2011 Mathematics Subject Classifications 51E15 To Professor Spyros Magliveras on his 70th birthday Abstract In this study we show the existence of subplanes of order 3 in Hughes planes of order q2 where q is a prime power and q 5 mod 6 . We further show that there exist finite partial linear spaces which cannot embed in any Hughes plane. 1 Introduction L. Puccio and M. J. de Resmini 5 showed that subplanes of order 3 exist in the Hughes plane of order 25. We refer always to the ordinary Hughes planes equivalently all our nearfields are regular. Computations of the second author 2 show that among the known projective planes of order 25 including 99 planes up to isomorphism duality exactly four have subplanes of order 3. These four planes are the ordinary Hughes plane and three closely related planes. Recently Caliskan and Magliveras 1 showed that there are exactly 2 orbits on subplanes of order 3 in the Hughes plane of order 121. In this study we show that every Hughes plane of order q2 where q is a prime power and q 5 mod 6 has subplanes of order 3. We begin with the construction of the Hughes plane H q2 of order q2 q an odd prime power as given by Rosati 6 and Zappa 9 . Throughout this paper K denotes a finite field of order q2 and F its subfield of order q where q is an odd prime power. For any 0 G K with 0 G F we have K F 0 and 1 0 is a basis for K over F. We will always choose 0 such that 02 d G F where d is a nonsquare in F. We now define the regular nearfield N of order q2 where N has the same elements as K and the same addition. However multiplication in N is defined as follows a o b ab if a is a square in K and a o b abq otherwise.

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