Axioms for fanos geometry minnesota state university. The fano plane is an example of an n3 configuration, that is, a set of n points and n lines with three points on each line and three lines through each point. Obviously, autu contains the identity map and hence is not empty. In 5, 17the authors eliminated all but one nontrivial group as possible automorphism groups of a binary qanalog of the fano plane, so that the automorphism group is known to be at most of. In this work we concentrate on the fano plane as an example remarking that our results can be possibly generalised. When the geometry has elements termed lines, such as for projective planes, the automorphism is alternatively called a collineation. The set of all automorphisms of u is denoted by autu. An anti ag in the fano plane is a nonincident pointline pair. Opposition diagrams for automorphisms of large spherical. There are 30 possible ways of arranging 7 objects into 7 triads such that each pair of objects belongs to exactly 1 triad. Finite projective geometries and linear codes tom edgar advisor. As a result 4, theorem 1, its automorphism group is at most of order 4, and up to conjugacy in gl7. We constructed a projective plane by taking a 3dimensional vector space v over a field f and. Automorphism group of fano plane mathematics stack exchange.
Produce two decks of cards, one bluebacked, one redbacked, and explain that the volunteer is going to determine some random placement of bluebacked cards at the points of the figure, which will. Orbits definitions counting the known planes automorphisms of classical planes. The nal construction we give relates to the fano plane. The mathieu groups and their geometries eindhoven university. We prove that if there exist both type j 1 and j 2 simplices.
These are organized into flipped and nonflipped pairs associated with the 240 assigned particles to e8. The question if there exists a qanalog of the fano plane is open since it was first posed in 1972. Finite permutation groups and finite classical groups 49 figure 4. Pdf there are two types of quadrangles in a protective plane, fano. In 5,17 the authors eliminated all but one nontrivial group as possible automorphism groups of a binary qanalog of the fano plane, so that the automorphism group is known to be at most of. At the outset, the top eight cards of the bluebacked deck are. Fano plane and petersen graph see also the background pattern for this page. The simplest example is given by the blowup of along a plane cubic, that is, a smooth fano threefold with. The 7,3,1difference set 0,1,3 produces the fano plane with blocks 0, 124, 235, 346, 045, 156, 026 has the automorphism.
Heawood graph, which is a way to encode the fano plane. An automorphism is an isomorphism of a rank two geometry to itself. Here, it is shown with theoretical and computational methods that the order of the automorphism group of a binary qanalog of the fano plane is either trivial or of order 2. Every line has exactly three points incident to it. Difference sets mathematical and statistical sciences. A binary qanalog of the fano plane is either rigid or its automorphism group is cyclic of order 2, 3 or 4.
On qanalogs of the fano plane universitat bayreuth. The smallest projective plane is p2f 2, where f2 is the. The composition of any two maps in autu is again an element of autu. The automorphism group of the fano plane is order 168. Fano threefolds with infinite automorphism groups iopscience.
In other words, it consists of lines through the origin in the vector space. The smallest set of admissible parameters of a qanalog of a steiner system is s 2 2, 3, 7. Remark that the image depicts the fano plane, a delightful mathematical entity representing the smallest finite geometry, of which you are a big fan. Charles weibel is professor of mathematics at rutgers university.
The order of the automorphism group of a binary qanalog of. We show that only 28 of these labelings of the fano plane are nonequivalent, which leads us to consider the automorphism group of the octonions. Second problem set, mostly on square designs and intersection triangles. Letters, the existence of zariski dense orbits for endomorphisms of projective surfaces with an appendix in collaboration with thomas tucker, 2019, 65pp, pdf. The existence of such a steiner systemknown as a binary qanalog of the fano planeis still open. Automorphisms of fano contact manifolds simon salamon special holonomy in geometry, analysis, and physics duke university, april 2018 reporting on results of j. At a convention a few years ago, the social golfer problem came up in a dinner conversation.
Fanos geometry handout axioms for fanos geometry undefined terms. In mathematics, the projective special linear group psl2, 7, isomorphic to gl3, 2, is a finite simple group that has important applications in algebra, geometry, and number theory. As a result 4, theorem 1, its automorphism group is at. An automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. An isotopy of an algebra is a triple of bijective linear maps a, b, c such that if xy z then axby cz. These 30 sets of triples form a single orbit under the action. When one uses the fano plane, everything aligns perfectly. In this paper we investigate the automorphism group of a putative binary qanalog of the fano plane. As referenced above, gpsl2,7 has 168 elements and is the automorphism group of the klein quartic as well as the symmetry group of the fano plane. Recall that credit wikipedia a projective plane consists of a set of lines, a set of points, and a relation between points and lines called incidence, having the following properties. The fano plane has an automorphism group of order 168.
The fano plane notice that, under right multiplication on both sides by k, 3. The automorphism group of plane algebraic curves with. Opposition diagrams for automorphisms of large spherical buildings james parkinson hendrik van maldeghem october 2, 2018 abstract let be an automorphism of a thick irreducible spherical building of rank at least 3 with no fano plane residues. As a conclusion, it is either rigid or its automorphism group is cyclic of order 2, 3 or 4. On the automorphism group of the binary qanalog of the. These are organized into flipped and nonflipped pairs associated with the 240 assigned particles to e8 vertices sorted by fano plane index or fpi.
Octonions have seven imaginary units whose multiplication table can be encoded using the fano plane mnemonic, shown here as a directed graph. On the automorphism group of the binary qanalog of the fano plane michael braun 1, michael kiermaier 2, and anamari naki c. If is a smooth fano threefold and its automorphism group is infinite, then is rational. Honors college thesis the geometry of the octonionic. How to prove that the fano plane is the smallest finite. You can use a proof similar to the above to show the simplicity of this automorphism group.
In this article, the automorphism group of a putative binary qanalog of the fano plane is investigated by a combination of theoretical and computational methods. From biplanes to the klein quartic and the buckyball. In particular, we count how many ways the fano plane can be labeled as the octonionic multiplication table, all corresponding to a specified octonion algebra. We will then turn to the important concept of collineations or automorphisms of projective planes. The automorphism group of plane algebraic curves with singer. Open problems concerning automorphism groups of projective. In particular, we count how many ways the fano plane can be labeled as the octonionic multiplication table, all corresponding to a speci ed octonion algebra. The order of the automorphism group of a binary qanalog of the fano is.
The product of two distinct units equals the unique unit such that the three units form three immediately connected vertices of the graph, multiplied by the signature of the permutation that orders the. In general, a projective plane has order n if each line. There is a unique projective plane of order 2 which is known as the fano plane. The fano plane, a 7 3 configuration, is unique and is the smallest such configuration. The codegree density of the fano plane dhruv mubayi. The nal nda over the real numbers is the octonions, o. Equilateral triangles and the fano plane institut camille jordan. The geometry of the octonionic multiplication table. Suppose p is a projective space of dimension d, and let q. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as. Under composition, the set of automorphisms of a graph forms what algbraists call a group. We construct embeddings using rotation systems see 14 and analyze the action of fano planes automorphism group pgl3. With baohua fu, wenhao ou, on fano manifolds of picard number one with big automorphism groups, 2018, 10pp, arxiv. The octonions are spanned by one identity element, 1, and 7 other basis elements.
In particular, i need a vector graphic of it and i would prefer to use the tikz package. Chapter 2 schwarz lemma and automorphisms of the disk. Since every such line contains a single nonzero element, we can also think of the fano plane as consisting of the seven nonzero elements of. It is the automorphism group of the klein quartic as well as the symmetry group of the fano plane. Some number of golfers desire to golf in foursomes over a period of weeks, without any two people being in the same foursome twice. If we view these anti ags as vertices of a graph, with p 1. On the automorphism group of a binary qanalog of the fano. On the automorphism group of the binary qanalog of the fano. It has 7 points and 7 lines, and is often called the fano plane, having been discovered in 1892 by gino fano fano. Apr 04, 2016 focus onbinary qanalog of the fano plane sts27. The automorphism group acts on the 7 lines of the fano plane. I am pleased to announce the availability of splitfano. Opposition diagrams for automorphisms of large spherical buildings james parkinson hendrik van maldeghem december 17, 2017 abstract let be an automorphism of a thick irreducible spherical building of rank at least 3 with no fano plane residues.
For a putative binary qanalog of the fano plane all automorphisms of order greater than 4 had been excluded previously. First problem set, exploring the fano plane and generalizations and petersen graph from the introductory handout. We will return to this idea later when we discuss the imbedding of the fano plane in kleins surface. The product of two distinct units equals the unique unit such that the three units form three immediately connected vertices of the graph, multiplied by the. Given any two distinct points, there is exactly one line i. The fano plane is the projective plane over the 2element field.
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