Algebraic structures related with finite projective planes Kuznetsov Eugene (Chisinau, MOLDOVA) Institute of Mathematics and Computer Science ”Vladimir Andrunachievici” Chisinau, MOLDOVA E-mail: kuznet1964@mail.ru International Algebraic Conference, Hungary, Budapest, July 7-13, 2019
The contents 1. Definitions. 2. Prime Power Conjecture (PPC). 3. Coordinatization of finite projective plane. 4. Sharply 2-transitive set of permutations of degree n . 5. Transversals and generalized transversals in a group to its subgroup. 6. Transversals with isotopic and isomorphic transversal operations. 7. Sharply 2-transitive set of permutations degree n and loop transversal in S n to St ab ( S n ). 8. Group case - a finite sharply 2-transitive group of permutations degree n . 9. A scheme of my proof of Hall’s Theorem.
The contents 10. An idea of generalization of above mentioned proof on a loop case. 11. Transversals in a loop to its suitable subloop. 12. Structural theorems. 13. An uniqueness of the loop transversal consists of fixed-point-free permutations. 14. A scheme of a proof of generalization of Hall’s Theorem on a loop case. 15. Properties of the loop transversal consists of fixed-point-free permutations. 16. Proof of PPC. 17. References.
Definitions Definition 1 A projective plane is an incidence structure < X , L , I > which satisfies the following axioms: 1 Given any two distinct points from X there exists just one line from L incident with both of them; 2 Given any two distinct lines from L there exists just one point from X incident with both of them; 3 There exist four points such that a line incident with any two of them is not incident with either of the remaining two. Theorem 2 A finite projective plane P may be formally defined as a set of n 2 + n + 1 points with the properties that: 1 Any two points determine a line, 2 Any two lines determine a point, 3 Every point has n + 1 lines on it, 4 Every line contains n + 1 points. The number n is called an order of the plane P .
Prime Power Conjecture (PPC) Using the vector space construction with finite fields there exists a projective plane of order n = p m , for each prime power p m . In fact, for all known finite projective planes, the order n is a prime power. The existence of finite projective planes of other orders is an open question. There exists the famous Prime Power Conjecture for projective planes. Problem 1 . ( Prime Power Conjecture (PPC) for projective planes ) Finite projective planes have prime power order.
Coordinatization of finite projective plane In authors survey [Kuznetsov1995.1] it was demonstrated the correlations in the following scheme: Projective plane A set of cell permutations of order n of DK -ternar of order n � ր � A sharply 2-transitive set of DK -ternar of order n ↔ permutations of degree n � � The loop of pairs of A loop transversal ↔ DK -ternar of order n in S n to St ab ( S n ) � � The loop of order n ( n − 1) A sharply 2-transitive loop of with some conditions on ↔ permutations of degree n cosets by two subloops
Coordinatization of finite projective plane Definition 3 [Kuznetsov1987] A system < E , ( x , t , y ) , 0 , 1 > is called a finite DK-ternar (e.g. a set E with ternary operation ( x , t , y ) and distinguished elements 0 , 1 ∈ E ), if the following conditions hold: 1 ( x , 0 , y ) = x ; 2 ( x , 1 , y ) = y ; 3 ( x , t , x ) = x ; 4 (0 , t , 1) = 0; 5 If a , b , c and d are arbitrary elements from E and a � = b , then the system � ( x , a , y ) = c ( x , b , y ) = d has an unique solution in E × E .
Coordinatization of finite projective plane Lemma 4 [Kuznetsov1987] Let π be a projective plane. It is possible to introduce coordinates ( a , b ) , ( m ) , ( ∞ ) for points and [ a , b ] , [ m ] , [ ∞ ] for lines from π (where a , b , m ∈ E, E is some set with distinguished elements 0 and 1), such that for operation ( x , t , y ) , where def ( x , t , y ) = z ⇔ ( x , y ) ∈ [ t , z ] , the system < E , ( x , t , y ) , 0 , 1 > is a DK-ternar. Lemma 5 Let the system < E , ( x , t , y ) , 0 , 1 > be a DK-ternar. Let a , b be arbitrary elements from E and a � = b. Then any unary operation α a , b ( t ) = ( a , t , b ) is a permutation on the set E. The permutations from Lemma 5 are called cell permutations .
Sharply 2-transitive set of permutations degree n Definition 6 A set M of permutations on a set X is called sharply 2-transitive , if for any two pairs ( a , b ) and ( c , d ) of different elements from X there exists an unique permutation α ∈ M satisfying the following conditions α ( a ) = c , α ( b ) = d . Lemma 7 [Kuznetsov1987] Cell permutations satisfy of the following conditions: 1 All cell permutations are distinct; 2 There exists fixed-point-free permutation ν on E such that we can describe all fixed-point-free cell permutations (with the identity cell permutation α 0 , 1 ( t ) ) by the following form: α ( t ) = ( a , t , ν ( a )) , ( ν (0) = 1) . 3 The set M of all cell permutations is sharply 2-transitive on the set E.
Transversals and generalized transversals in a group to its subgroup Definition 8 (Kuznetsov2016) Let G be a group and H be its subgroup. Let { H i } i ∈ E be the set of all left cosets in G to H . A set T = { t i } i ∈ E of representativities of the left (right) cosets (by one from each coset H i , i.e. t i ∈ H i ) is called a left generalized transversal in G to H (see also [Pflugfelder1991]). Definition 9 (Baer1939) A left generalized transversal T = { t i } i ∈ E in G to H which satisfy the following conditions: t i 0 = e for some i 0 ∈ E and H = H 1 , is usually called a left transversal in G to H . Definition 10 Let T = { t i } i ∈ E be a left generalized transversal in G to H . Define the following operation on the set E : ( T ) x · y = z ⇔ t x t y = t z h , h ∈ H .
Transversals and generalized transversals in a group to its subgroup Theorem 11 For an arbitrary left generalized transversal T = { t i } i ∈ E in G to H the following statements are true: 1 There exists an element a 0 ∈ E such that the system � � ( T ) · , a 0 E , is a left quasigroup with right unit a 0 . 2 If T = { t i } i ∈ E is a left transversal in G to H, then the system � � ( T ) E , · , 1 is a left loop with unit 1 .
Transversals with isotopic and isomorphic transversal operations Theorem 12 (Kuznetsov2016) For an arbitrary left generalized transversal T = { t i } i ∈ E in G to H the following statements are true: 1 If P = { p i } i ∈ E is a left generalized transversal in G to H such that for every x ∈ E: P = Th 0 , p x ′ = t x h 0 , where h 0 ∈ H is an arbitrary fixed element, then the transversal operation � � � � ( P ) ( T ) E , · is isotopic to the transversal operation E , · . 2 If S = { s i } i ∈ E is a left generalized transversal in G to H such that for every x ∈ E: S = π T , s x ′ = π t x , where π ∈ G is an arbitrary fixed element, then the transversal operation � � � � ( S ) ( T ) E , · is isotopic to the transversal operation E , · .
Transversals with isotopic and isomorphic transversal operations Lemma 13 (Kuznetsov2010.2) Let T = { t x } x ∈ E be a fixed loop transversal in G to H and h 0 ∈ N St 1 ( S E ) ( H ) . Define the set of permutations: p x ′ def = h − 1 0 t x h 0 ∀ x ∈ E . Then 1 P = { p x ′ } x ′ ∈ E is a loop transversal in G to H; � � � � ( P ) ( T ) 2 The transversal operations E , · , 1 and E , · , 1 are isomorphic, and the isomorphism is set up by the mapping ϕ ( x ) = h 0 ( x ) . Lemma 14 (Kuznetsov2014) Let T = { t x } x ∈ E be a fixed loop transversal in G to H. Let h 0 ∈ N St 1 ( S E ) ( H ) be an element such that: t x ′ def = h − 1 0 t x h 0 ∀ x ∈ E . � � ( T ) Then ϕ ≡ h 0 ∈ Aut E , · , 1 .
Sharply 2-transitive set of permutations degree n and loop transversal in S n to St ab ( S n ) Lemma 15 (Kuznetsov1994.2) Let E be a finite set and | E | = n. The following conditions are equivalent: 1 A set T is a loop transversal in S n to St a , b ( S n ) , where a , b ∈ E are arbitrary fixed distinct elements; 2 A set T is a sharply 2-transitive set of permutations on E; 3 A set T is a sharply 2-transitive permutation loop on E.
Group case - a finite sharply 2-transitive group of permutations degree n In the theory of finite multiply transitive permutation groups the following M. Hall’s theorem is well-known [Hall1962]. Theorem 16 Let G be a sharply 2 -transitive permutation group on a finite set of symbols E, i.e. 1 G is a 2 -transitive permutation group on E; 2 only identity permutation id fixes two symbols from the set E. Then 1 the identity permutation id with the set of all fixed-point-free permutations from the group G forms a transitive invariant subgroup A in the group G; 2 group G is isomorphic to the group of linear transformations G K = { α | α ( x ) = x · a + b , a , b ∈ E , a � = 0 } of a some near-field K = � E , + , · , 0 , 1 � .
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