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On projected planes
 Date Issued:
 2010
 Summary:
 This work was motivated by the wellknown question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p < 11, there is only the pappian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are nondesarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras: "Begin with a nondesarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is nondesarguesian." In this manuscript we use a grouptheoretic methodology to determine the subplane structures of some nondesarguesian planes. In particular, we determine orbit representatives of all proper Qsubplanes both of a VeblenWedderburn (VW) plane of order 121 and of the Hughes plane of order 121, under their full collineation groups. In PI, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Sigma , there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408; 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some nondesarguesian planes, such as the Hall plane of order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups.
Title:  On projected planes. 
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Name(s): 
Caliskan, Cafer. Charles E. Schmidt College of Science Department of Mathematical Sciences 

Type of Resource:  text  
Genre:  Electronic Thesis Or Dissertation  
Date Issued:  2010  
Publisher:  Florida Atlantic University  
Physical Form:  electronic  
Extent:  x, 60 p. : ill.  
Language(s):  English  
Summary:  This work was motivated by the wellknown question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p < 11, there is only the pappian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are nondesarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras: "Begin with a nondesarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is nondesarguesian." In this manuscript we use a grouptheoretic methodology to determine the subplane structures of some nondesarguesian planes. In particular, we determine orbit representatives of all proper Qsubplanes both of a VeblenWedderburn (VW) plane of order 121 and of the Hughes plane of order 121, under their full collineation groups. In PI, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Sigma , there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408; 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some nondesarguesian planes, such as the Hall plane of order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups.  
Identifier:  610569074 (oclc), 1927609 (digitool), FADT1927609 (IID), fau:2958 (fedora)  
Note(s): 
by Cafer Caliskan. Thesis (Ph.D.)Florida Atlantic University, 2010. Includes bibliography. Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web. 

Subject(s): 
Projected planes Combinatorial designs and configurations Surfaces, Algebraic Manifolds (Mathematics) 

Persistent Link to This Record:  http://purl.flvc.org/FAU/1927609  
Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU 