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| In [[combinatorial theory]], a '''generalized polygon''' is an incidence structure introduced by [[Jacques Tits]]. Generalized polygons encompass as special cases [[projective plane]]s (generalized triangles, ''n'' = 3) and [[generalized quadrangle]]s (''n'' = 4). Many generalized polygons arise from [[groups of Lie type]], but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the ''[[Ruth Moufang|Moufang]] property'' have been completely classified by Tits and Weiss. Every generalized polygon is also a [[near polygon]].
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| ==Definition==
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| A generalized ''2''-gon (or a digon) is a [[partial linear space]] where each point is incident to each line. For ''n > 3'' a generalized ''n''-gon is an [[incidence structure]] (<math>P,L,I</math>), where <math>P</math> is the set of points, <math>L</math> is the set of lines and <math>I\subseteq P\times L</math> is the [[incidence relation]], such that:
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| * It is a partial linear space.
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| * It has no ordinary ''m''-gons as subgeometry for ''2 < m < n''.
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| * It has ordinary ''n''-gon as a subgeometry.
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| * For any <math> \{A_1, A_2\} \subseteq P \cup L </math> there exists a subgeometry (<math> P', L', I' </math>) isomorphic to an ordinary ''n''-gon such that <math>\{A_1, A_2\} \subseteq P' \cup L' </math>.
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| An equivalent but sometimes simpler way to express these conditions is: consider the [[bipartite graph|bipartite]] ''incidence graph'' with the vertex set <math>P \cup L</math> and the edges connecting the incident pairs of points and lines.
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| * The [[girth (graph theory)|girth]] of the incidence graph is twice the [[diameter (graph theory)|diameter]] 'n' of the incidence graph.
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| A generalized polygon is of order ''(s,t)'' if:
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| * all vertices of the incidence graph corresponding to the elements of <math>L</math> have the same degree ''s'' + 1 for some natural number ''s''; in other words, every line contains exactly ''s'' + 1 points,
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| * all vertices of the incidence graph corresponding to the elements of <math>P</math> have the same degree ''t'' + 1 for some natural number ''t''; in other words, every point lies on exactly ''t'' + 1 lines.
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| We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.
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| The dual of a generalized ''n''-gon (<math>P,L,I</math>), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the [[inverse relation]] of <math>I</math>. It can easily be shown that this is again a generalized ''n''-gon.
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| ==Examples==
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| * A generalized digon is a [[complete bipartite graph]] K<sub>s+1,t+1</sub>.
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| * For any natural ''n'' ≥ 3, consider the boundary of the ordinary [[polygon]] with ''n'' sides. Declare the vertices of the polygon to be the points and the sides to be the lines, with set inclusion as the incidence relation. This results in a generalized ''n''-gon with ''s'' = ''t'' = 1.
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| * For each [[group of Lie type]] ''G'' of rank 2 there is an associated generalized ''n''-gon ''X'' with ''n'' equal to 3, 4, 6 or 8 such that ''G'' acts transitively on the set of flags of ''X''. In the finite case, for ''n=6'', one obtains the Split Cayley hexagon of order ''(q,q)'' for [[List_of_finite_simple_groups#G2.28q.29_Chevalley_groups|''G''<sub>2</sub>(''q'')]] and the twisted triality hexagon of order ''(q<sup>3</sup>,q)'' for [[List_of_finite_simple_groups#3D4.28q3.29_Steinberg_groups|<sup>3</sup>''D''<sub>4</sub>(''q''<sup>3</sup>)]], and for ''n=8'', one obtains the Ree-Tits octagon of order ''(q,q<sup>2</sup>)'' for [[List_of_finite_simple_groups#3D4.28q3.29_Steinberg_groups|<sup>2</sup>''F''<sub>4</sub>(q)]] with ''q=2<sup>2''n''+1</sup>''. Up to duality, these are the only known thick finite generalized hexagons or octagons.
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| == Feit-Higman theorem ==
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| [[Walter Feit]] and [[Graham Higman]] proved that ''finite'' generalized ''n''-gons with
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| ''s'' ≥ 2, ''t'' ≥ 2 can exist only for the following values of ''n'':
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| :2, 3, 4, 6 or 8.
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| Moreover,
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| * If ''n'' = 2, the structure is a complete bipartite graph.
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| * If ''n'' = 3, the structure is a finite [[projective plane]], and ''s'' = ''t''.
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| * If ''n'' = 4, the structure is a finite [[generalized quadrangle]], and ''t''<sup>1/2</sup> ≤ ''s'' ≤ ''t''<sup>2</sup>.
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| * If ''n'' = 6, then ''st'' is a [[Square number|square]], and ''t''<sup>1/3</sup> ≤ ''s'' ≤ ''t''<sup>3</sup>.
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| * If ''n'' = 8, then ''2st'' is a square, and ''t''<sup>1/2</sup> ≤ ''s'' ≤ ''t''<sup>2</sup>.
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| * If ''s'' or ''t'' is allowed to be 1 and the structure is not the ordinary ''n''-gon then besides the values of ''n'' already listed, only ''n'' = 12 may be possible.
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| If ''s'' and ''t'' are both infinite then generalized polygons exist for each ''n'' greater or equal to 2. It is unknown whether or not there exist generalized polygons with one of the parameters finite and the other infinite (these cases are called ''semi-finite'').
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| == See also ==
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| * [[Building (mathematics)]]
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| * [[(B, N) pair]]
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| * [[Ree group]]
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| * [[Moufang plane]]
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| * [[Near polygon]]
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| ==References==
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| *{{citation
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| | last1 = Godsil | first1 = Chris | author1-link = Chris Godsil
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| | last2 = Royle | first2 = Gordon | author2-link = Gordon Royle
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| | doi = 10.1007/978-1-4613-0163-9
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| | isbn = 0-387-95220-9
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| | location = New York
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| | mr = 1829620
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| | publisher = Springer-Verlag
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| | series = Graduate Texts in Mathematics
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| | title = Algebraic Graph Theory
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| | volume = 207
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| | year = 2001}}.
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| *{{citation
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| | last1 = Feit | first1 = Walter | author1-link = Walter Feit
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| | last2 = Higman | first2 = Graham | author2-link = Graham Higman
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| | doi = 10.1016/0021-8693(64)90028-6
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| | journal = Journal of Algebra
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| | mr = 0170955
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| | pages = 114–131
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| | title = The nonexistence of certain generalized polygons
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| | volume = 1
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| | year = 1964}}.
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| *{{citation
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| | last = van Maldeghem | first = Hendrik
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| | doi = 10.1007/978-3-0348-0271-0
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| | isbn = 3-7643-5864-5
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| | location = Basel
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| | mr = 1725957
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| | publisher = Birkhäuser Verlag
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| | series = Monographs in Mathematics
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| | title = Generalized polygons
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| | volume = 93
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| | year = 1998}}.
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| *{{citation
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| | last = Stanton | first = Dennis
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| | doi = 10.1016/0097-3165(83)90036-5
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| | issue = 1
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| | journal = [[Journal of Combinatorial Theory]]
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| | mr = 685208
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| | pages = 15–27
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| | series = Series A
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| | title = Generalized ''n''-gons and Chebychev polynomials
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| | volume = 34
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| | year = 1983}}.
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| *{{citation
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| | last1 = Tits | first1 = Jacques | author1-link = Jacques Tits
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| | last2 = Weiss | first2 = Richard M.
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| | isbn = 3-540-43714-2
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| | location = Berlin
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| | mr = 1938841
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| | publisher = Springer-Verlag
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| | series = Springer Monographs in Mathematics
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| | title = Moufang polygons
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| | year = 2002}}.
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| [[Category:Group theory]]
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| [[Category:Incidence geometry]]
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37 yr old Pig Farmer Reinaldo Deniston from Saint-Bruno, really loves going to movies, como ganhar dinheiro na internet and calligraphy. Has toured ever since childhood and has gone to many places, for example Anjar.