k –Nets in a Projective Plane over a Field Nicola Pace 1 (ICMC, University of S˜ ao Paulo) joint work with G.Korchmaros (Univ. della Basilicata, Italy) and G.Nagy (Univ. of Szeged, Hungary) Special Days on Combinatorial Constructions using Finite Fields Linz, December 5–6, 2013 1 Supported by FAPESP (Funda¸ c˜ ao de Amparo a Pesquisa do Estado de S˜ ao Paulo), procs no. 12/03526-0. Nicola Pace (ICMC, University of S˜ ao Paulo) joint work with G.Korchmaros (Univ. della Basilicata, Italy) and G.Nagy (Univ. of Sz k –nets
Outline 3-nets (in particular, 3-nets realizing groups) 2 / 34
Outline 3-nets (in particular, 3-nets realizing groups) Examples: Algebraic 3-nets Tetrahedron type 3-nets 2 / 34
Outline 3-nets (in particular, 3-nets realizing groups) Examples: Algebraic 3-nets Tetrahedron type 3-nets Classification of 3-nets realizing group 2 / 34
Outline 3-nets (in particular, 3-nets realizing groups) Examples: Algebraic 3-nets Tetrahedron type 3-nets Classification of 3-nets realizing group Some recent result on k-nets, k ≥ 4. 2 / 34
Projective Plane PG ( 2 , K ) Let K be a field Points: P : ( x , y , z ) ∈ K × K × K , ( x , y , z ) � = (0 , 0 , 0) ( x , y , z ) ∼ ( kx , ky , kz ) , for k ∈ K \ { 0 } Lines: ℓ : aX + bY + cZ = 0 , a , b , c ∈ K , ( a , b , c ) � = (0 , 0 , 0) Incidence Relation I : P I ℓ ⇐ ⇒ ax + by + cz = 0 3 / 34
Projective Planes Definition A projective plane P is a set of points and lines, together with an incidence relation between the points and the lines such that Any two distinct points are incident with a unique line. 1 Any two distinct lines are incident with a unique point. 2 There exists four points no three of which are incident with 3 one line. 4 / 34
Projective Planes Definition A projective plane P is a set of points and lines, together with an incidence relation between the points and the lines such that Any two distinct points are incident with a unique line. 1 Any two distinct lines are incident with a unique point. 2 There exists four points no three of which are incident with 3 one line. Remark PG (2 , K ) is a very particular projective plane. 4 / 34
Projective Planes Definition A projective plane P is a set of points and lines, together with an incidence relation between the points and the lines such that Any two distinct points are incident with a unique line. 1 Any two distinct lines are incident with a unique point. 2 There exists four points no three of which are incident with 3 one line. Remark PG (2 , K ) is a very particular projective plane. ... with a very special property. 4 / 34
Desargues’ Theorem [the special property] 5 / 34
Fano Plane: PG ( 2 , F 2 ) (source: http://home.wlu.edu/ ∼ mcraea/) 6 / 34
PG ( 2 , F 3 ) (source: http://home.wlu.edu/ ∼ mcraea/) 7 / 34
3-nets Definition A 3 -net in PG (2 , K ) is a pair ( A , X ) where A is a finite set of lines partitioned into 3 subsets A = A 1 ∪ A 2 ∪ A 3 and X is a finite set of points subject to the following conditions: for every i � = j and every ℓ ∈ A i , ℓ ′ ∈ A j , we have ℓ ∩ ℓ ′ ∈ X for every X ∈ X and every i (i ∈ { 1 , 2 , 3 } ) there exists a unique line ℓ ∈ A i passing through X. 8 / 34
3-nets Definition A 3 -net in PG (2 , K ) is a pair ( A , X ) where A is a finite set of lines partitioned into 3 subsets A = A 1 ∪ A 2 ∪ A 3 and X is a finite set of points subject to the following conditions: for every i � = j and every ℓ ∈ A i , ℓ ′ ∈ A j , we have ℓ ∩ ℓ ′ ∈ X for every X ∈ X and every i (i ∈ { 1 , 2 , 3 } ) there exists a unique line ℓ ∈ A i passing through X. Note: |A 1 | = |A 2 | = |A 3 | = n , |X| = n 2 ( n is the order of the 3-net) 8 / 34
3-nets 9 / 34
3-nets 9 / 34
3-nets 9 / 34
(dual) 3-nets points ↔ lines lines ↔ points 10 / 34
(dual) 3-nets points ↔ lines lines ↔ points Definition A 3 -net in PG (2 , K ) is a pair ( A , X ) where A is a finite set of lines partitioned into 3 subsets A = A 1 ∪ A 2 ∪ A 3 and X is a finite set of points subject to the following conditions: for every i � = j and every ℓ ∈ A i , ℓ ′ ∈ A j , we have ℓ ∩ ℓ ′ ∈ X for every X ∈ X and every i (i ∈ { 1 , 2 , 3 } ) there exists a unique line ℓ ∈ A i passing through X. 10 / 34
(dual) 3-nets points ↔ lines lines ↔ points Definition A (dual) 3 -net in PG (2 , K ) is a pair ( A , X ) where A is a finite set of points partitioned into 3 subsets A = A 1 ∪ A 2 ∪ A 3 and X is a finite set of lines subject to the following conditions: for every i � = j and every P ∈ A i , P ′ ∈ A j , we have that the 1 line PP ′ ∈ X for every ω ∈ X and every i (i ∈ { 1 , 2 , 3 } ) there exists a 2 unique point P ∈ A i passing through ω . 10 / 34
(dual) 3-nets 11 / 34
(dual) 3-nets 11 / 34
(dual) 3-nets 11 / 34
(dual) 3-nets, quasigroups, loops 12 / 34
(dual) 3-nets, quasigroups, loops * 1 2 3 . . . n . . 1 . 2 . . . . . . 4 3 ⇒ . . . . . . n 12 / 34
(dual) 3-nets, quasigroups, loops Definition A quasigroup ( Q , ∗ ) is a set Q with a binary operation ∗ , such that for each a , b ∈ Q, there exist unique elements x and y in Q such that: a ∗ x = b , y ∗ a = b . * 1 2 3 . . . n . . 1 . 2 . . . . . . 4 3 ⇒ . . . . . . n 12 / 34
(dual) 3-nets, quasigroups, loops Definition A quasigroup ( Q , ∗ ) is a set Q with a binary operation ∗ , such that for each a , b ∈ Q, there exist unique elements x and y in Q such that: a ∗ x = b , y ∗ a = b . Definition A loop is a quasigroup with an identity element e such that: x ∗ e = x = e ∗ x for all x in Q . 12 / 34
(dual) 3-nets, quasigroups, loops 12 / 34
(dual) 3-nets, quasigroups, loops 12 / 34
(dual) 3-nets, quasigroups, loops 12 / 34
(dual) 3-nets realizing groups A (dual) 3-net is said to realize a group ( G , · ) when it is coordinatized by G: if A 1 , A 2 , A 3 are the classes, there exists a triple of bijective maps from G to ( A 1 , A 2 , A 3 ), say α : G → A 1 , β : G → A 2 , γ : G → A 3 such that a · b = c if and only if α ( a ), β ( b ), γ ( c ) are three collinear points, for any a , b , c ∈ G . 13 / 34
Main Goal Problem Classify dual 3-nets! 14 / 34
Main Goal Problem Classify dual 3-nets! Comment Easily stated but too a general problem 14 / 34
Main Goal Problem Classify dual 3-nets! Comment Easily stated but too a general problem Question: Which groups can be realized? 14 / 34
Main Goal Problem Classify dual 3-nets! Comment Easily stated but too a general problem Question: Which groups can be realized? It depends on the characteristic of the field K ! If n ≥ 1, char ( K ) = 2 and K “large enough”, the group ( Z 2 ) n can be realized. If char ( K ) � = 2 , the group ( Z 2 ) 3 cannot be realized (Yuzvinsky, 2003). 14 / 34
Main Goal Problem Classify dual 3-nets! Comment Easily stated but too a general problem Question: Which groups can be realized? It depends on the characteristic of the field K ! If n ≥ 1, char ( K ) = 2 and K “large enough”, the group ( Z 2 ) n can be realized. If char ( K ) � = 2 , the group ( Z 2 ) 3 cannot be realized (Yuzvinsky, 2003). Some restrictions are needed. Our hypotheses are: (i) The 3-net (Λ 1 , Λ 2 , Λ 3 ) realizes a group G . (ii) p > n or p = 0, where | G | = n and p is the characteristic of the field. 14 / 34
Main Goal Problem Classify dual 3-nets! Comment Easily stated but too a general problem Question: Which groups can be realized? It depends on the characteristic of the field K ! If n ≥ 1, char ( K ) = 2 and K “large enough”, the group ( Z 2 ) n can be realized. If char ( K ) � = 2 , the group ( Z 2 ) 3 cannot be realized (Yuzvinsky, 2003). Some restrictions are needed. Our hypotheses are: (i) The 3-net (Λ 1 , Λ 2 , Λ 3 ) realizes a group G . (ii) p > n or p = 0, where | G | = n and p is the characteristic of the field. 14 / 34
Algebraic dual 3-nets Definition A dual 3-net (with n ≥ 4 ) is said to be algebraic if all its points lie on a (uniquely determined) plane cubic F , called the associated plane cubic. 15 / 34
Algebraic dual 3-nets Definition A dual 3-net (with n ≥ 4 ) is said to be algebraic if all its points lie on a (uniquely determined) plane cubic F , called the associated plane cubic. Algebraic dual 3-nets fall into subfamilies according as the plane cubic splits into three lines splits into an irreducible conic and a line is irreducible 15 / 34
Triangular dual 3-nets Theorem Every triangular dual 3-net realizes a cyclic group isomorphic to a multiplicative group of K . 16 / 34
Triangular dual 3-nets Theorem Every triangular dual 3-net realizes a cyclic group isomorphic to a multiplicative group of K . Proof: Assume the vertices of the triangle are O = (0 , 0 , 1) , X ∞ = (1 , 0 , 0) , Y ∞ = (0 , 1 , 0) . 16 / 34
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