[PPT] - Algebraic structures related with finite projective planes PowerPoint Presentation

SLIDE 1

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

SLIDE 2

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 Sn to Stab(Sn).

8. Group case - a finite sharply 2-transitive group of permutations

degree n.

9. A scheme of my proof of Hall’s Theorem.

SLIDE 3

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.

SLIDE 4

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 n2 + 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.

SLIDE 5

Prime Power Conjecture (PPC)

Using the vector space construction with finite fields there exists a projective plane of order n = pm, for each prime power pm. 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.

SLIDE 6

Coordinatization of finite projective plane

In authors survey [Kuznetsov1995.1] it was demonstrated the correlations in the following scheme: Projective plane

f order n

A set of cell permutations

f DK-ternar of order n
ր
DK-ternar of order n

↔ A sharply 2-transitive set of permutations of degree n

The loop of pairs of

DK-ternar of order n ↔ A loop transversal in Sn to Stab(Sn)

The loop of order n(n − 1)

with some conditions on cosets by two subloops ↔ A sharply 2-transitive loop of permutations of degree n

SLIDE 7

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.

SLIDE 8

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 (x, t, y) = z

def

⇔ (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 αa,b(t) = (a, t, b) is a permutation on the set E. The permutations from Lemma 5 are called cell permutations.

SLIDE 9

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 α0,1(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.

SLIDE 10

Transversals and generalized transversals in a group to its subgroup

Definition 8 (Kuznetsov2016)

Let G be a group and H be its subgroup. Let {Hi}i∈E be the set of all left cosets in G to H. A set T = {ti}i∈E of representativities of the left (right) cosets (by one from each coset Hi, i.e. ti ∈ Hi) is called a left generalized transversal in G to H (see also [Pflugfelder1991]).

Definition 9 (Baer1939)

A left generalized transversal T = {ti}i∈E in G to H which satisfy the following conditions: ti0 = e for some i0 ∈ E and H = H1, is usually called a left transversal in G to H.

Definition 10

Let T = {ti}i∈E be a left generalized transversal in G to H. Define the following operation on the set E: x

(T)

· y = z ⇔ txty = tzh, h ∈ H.

SLIDE 11

Transversals and generalized transversals in a group to its subgroup

Theorem 11

For an arbitrary left generalized transversal T = {ti}i∈E in G to H the following statements are true:

1 There exists an element a0 ∈ E such that the system

E,

(T)

· , a0

is a left quasigroup with right unit a0.

2 If T = {ti}i∈E is a left transversal in G to H, then the system

E,

(T)

· , 1

is a left loop with unit 1.

SLIDE 12

Transversals with isotopic and isomorphic transversal

perations

Theorem 12 (Kuznetsov2016)

For an arbitrary left generalized transversal T = {ti}i∈E in G to H the following statements are true:

1 If P = {pi}i∈E is a left generalized transversal in G to H such that for

every x ∈ E: P = Th0, px′ = txh0, where h0 ∈ H is an arbitrary fixed element, then the transversal operation

E,

(P)

·

is isotopic to the transversal operation
E,

(T)

·

.

2 If S = {si}i∈E is a left generalized transversal in G to H such that for

every x ∈ E: S = πT, sx′ = πtx, where π ∈ G is an arbitrary fixed element, then the transversal operation

E,

(S)

·

is isotopic to the transversal operation
E,

(T)

·

.

SLIDE 13

Transversals with isotopic and isomorphic transversal

perations

Lemma 13 (Kuznetsov2010.2)

Let T = {tx}x∈E be a fixed loop transversal in G to H and h0 ∈ NSt1(SE )(H). Define the set of permutations: px′ def = h−1

0 txh0

∀x ∈ E. Then

1 P = {px′}x′∈E is a loop transversal in G to H; 2 The transversal operations

E,

(P)

· , 1

and
E,

(T)

· , 1

are isomorphic,

and the isomorphism is set up by the mapping ϕ(x) = h0(x).

Lemma 14 (Kuznetsov2014)

Let T = {tx}x∈E be a fixed loop transversal in G to H. Let h0 ∈ NSt1(SE )(H) be an element such that: tx′ def = h−1

0 txh0

∀x ∈ E. Then ϕ ≡ h0 ∈ Aut

E,

(T)

· , 1

.

SLIDE 14

Sharply 2-transitive set of permutations degree n and loop transversal in Sn to Stab(Sn)

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 Sn to Sta,b(Sn), 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.

SLIDE 15

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

GK = {α | α(x) = x · a + b, a, b ∈ E, a = 0}

f a some near-field K = E, +, ·, 0, 1.

SLIDE 16

A scheme of my proof of Hall’s Theorem

Lemma 17 (Kuznetsov2005)

Let G be a sharply 2-transitive permutation group of degree n on a set E. Then the set T of all fixed-point-free permutations from G with the identity permutation id form an unique loop transversal in G to H = St0(G) = {α ∈ G|α(0) = 0, 0 ∈ E}.

Lemma 18

Let T be the loop transversal in G to H from the previous Lemma. Then the following propositions are true:

1 ∀a ∈ E the set Sa = T · t−1 a

(where ta ∈ T) be a loop transversal in G to H.

2 ∀a ∈ E the set Ra = ta ·T (where ta ∈ T) be a loop transversal in G to H.

Lemma 19

The transversal T from the previous Lemmas is a group transversal in G to H.

Lemma 20

1 The group transversal operation (T, ·) has a sharply transitive

automorphism group.

2 The group (T, ·) is a primary cyclic group of order n = pm, p is a prime

number.

SLIDE 17

An idea of generalization of above mentioned proof on a loop case

An idea of generalization of the proof from the previous slide on a loop case consists in a following scheme: to prove the loop analogs

f the Lemmas from the previous slide.This will enable us to

describe the internal structure of the loop transversal L in Sn to Stab(Sn) .

SLIDE 18

Transversals in a loop to its suitable subloop

Definition 21 (Pflugfelder1991)

Let L, · be a loop and R, · be its proper subloop. Then a left coset

f R is a set of the form

xR = {xr | r ∈ R}, and analogically for a right coset. The cosets in a loop to its subloop do not necessarily form a partition of the loop. This leads us to the following definition.

Definition 22 (Pflugfelder1991)

A loop L has a left (right) coset decomposition by its proper subloop R, if the left (right) cosets form a partition of the loop L, is equal for some set of indexes E

1

∪

i∈E(aiR) = L; 2

for every i, j ∈ E, i = j (aiR) ∩ (ajR) = ∅.

SLIDE 19

Transversals in a loop to its suitable subloop

Definition 23 (Kuznetsov2010.1)

(Left Condition A) A product at the left of an arbitrary element a

f a loop L by an arbitrary left coset of a loop L to its subloop R is

a left coset of the loop L to its subloop R too, is equal for every a, b ∈ L there exists an element c ∈ L such that a(bR) = cR.

Lemma 24

Let the left Condition A for a loop L and its subloop R is fulfilled. Then the following conditions hold:

1 Left cosets Ri form a left coset decomposition of the loop L; 2 If a loop L is finite, then the ”Lagrange property” takes place:

an order of the subloop R divides an order of the loop L.

SLIDE 20

Transversals in a loop to its suitable subloop

Definition 25

[Kuznetsov2010.1] Let L, ·, e be a loop, R, ·, e be its subloop and a left Condition A is fulfilled. Let {Rx}x∈E is a set of all left cosets in L to R that form a left coset decomposition of the loop L (E is a set of indexes). A set T = {tx}x∈E ⊂ L is called a left transversal in L to R if T is a complete set of representatives of the left cosets Rx in L to R, is equal there exists an unique element tx ∈ T such that tx ∈ Rx for every x ∈ E.

Lemma 26

The following conditions are equivalent:

1 A set T = {tx}x∈E is a left loop transversal in a loop L to its

subloop R;

2 A set ˆ

T = {ˆ tx}x∈E is a sharply transitive set of permutations in the group SE.

SLIDE 21

Structural theorems

The following lemma is an explanation of the necessity of the Condition A in the investigation of transversals in loops.

Lemma 27

[Kuznetsov2011.1] Let G be a group, H be its proper subgroup. Let K be a subgroup of group G such that H ⊆ K ⊂ G. If T = {ti}i∈E is a left transversal in G to H, then:

1 Let us denote

E1 = {x ∈ E | tx ∈ K}. Then a set T1 = T|K = {tj}j∈E1 is a left transversal in K to H;

2 It is true that

E1,

(T1)

· , 1

⊂
E,

(T)

· , 1

;

3 A left Condition A is fulfilled in the left loop

E,

(T)

· , 1

to its left

subloop

E1,

(T)

· , 1

: for every a, b ∈
E,

(T)

· , 1

and u ∈
E1,

(T)

· , 1

there exist c ∈
E,

(T)

· , 1

and

u1 ∈

E1,

(T)

· , 1

such that

a

(T)

· (b

(T)

· u) = c

(T)

· u1.

SLIDE 22

Structural theorems

Lemma 28

[Kuznetsov2011.1] Let groups H, K and G satisfy a following condition H ⊆ K ⊂ G, and let T ∗ = {tx}x∈E0 is a left transversal in G to K. Then it (as a set) can be always supplemented up to some left transversal T = {tx}x∈E in G to H.

Lemma 29

Let conditions of Lemma 28 hold. Let T ∗ = {tx}x∈E0 be a left transversal in G to K, and T = {tx}x∈E be a such left transversal in G to H, for which T ∗ ⊆ T (and E0 ⊆ E) (according to Lemma 28). According Lemma 27 a set T1 = T ∩ K = {tx}x∈E1 is a left transversal in K to H. Then the following statements are true:

1 elements of the set E0 form a left transversal in the left loop

E,

(T)

· , 1

to its left subloop
E1,

(T)

· , 1

.

2 The operations

E0,

(T ∗)

· , 1

and
E0,

(E0)

· , 1

are isomorphic (first
peration is a transversal operation corresponds to the left transversal T ∗

in G to K, and second operation is a transversal operation corresponds to a left transversal E0 in the left loop

E,

(T)

· , 1

to its left subloop
E1,

(T)

· , 1

.)

SLIDE 23

Structural theorems

Lemma 30

[Kuznetsov2011.1] Let L = (E, ·, 1) be a loop and R = (E1, ·, 1) be its proper subloop, and the left Condition A is fulfilled. Let T = {tx}x∈E0 be a left transversal in L to R, and H ⊆ St1(SL) be a permutation group such that:

1 LI(L) ⊆ H; 2 (L−1 h(u)h Luh−1) ∈ H

∀u ∈ L, h ∈ H (where La is a left translation in (E, ·, 1)) Then the following sentences are true:

1 It is possible to define correctly a semidirect product G = L ⋋ H. 2 A set

K = {(r, h) | r ∈ R, h ∈ H} is a subgroup of the group G, and moreover, H ⊂ K ⊆ G.

3 A set

T ∗ = {(tx, id) | tx ∈ T0, x ∈ E0} is a left transversal in the group G to its subgroup K.

4 The transversal operations

E0,

(T)

· , 1

and
E0,

(T ∗)

· , 1

(corresponding

to the left transversal T in the loop L to its subloop R, and left transversal in the group G to its subgroup K, respectively) coincide.

SLIDE 24

An uniqueness of the loop transversal consists of fixed-point-free permutations

Using the notion of a transversal in a loop to its subloop, we obtain a following generalization of Hall’s Theorem for the case of a sharply 2-transitive permutation loop.

Theorem 31 (Kuznetsov2005)

Let L be a sharply 2-transitive permutation loop on a finite set of symbols E, i.e.

1 L is a 2-transitive set of permutations on the finite set of symbols E; 2 permutations from the set L form a loop by some operation ”·”; 3 only identity permutation id fix two symbols from the set E.

Then

1 the identity permutation id with the set of all fixed-point-free

permutations from the loop L forms a transitive loop transversal A in the loop L to its proper subloop Ra, where Ra is a loop of all permutations from the loop L, which fix some fixed symbol a ∈ E;

2 this loop transversal A is an unique loop transversal in the loop L to

its proper subloop Ra, i.e. any other loop transversal T in the loop L to its proper subloop Ra coincide with the transversal T.

SLIDE 25

A scheme of a proof of generalization of Hall’s Theorem on a loop case

Theorem 32

Let a set A = {tx}x∈E be the loop transversal in the loop L to its subloop R0 from the previous Theorem. Then the following statements are true:

1 the set A∗ = {t∗ x = (tx, id)}x∈E is a loop transversal in the

multiplication group LM(L) to its subgroup K = R0 ⋋ LI(L);

2 ∀a ∈ E the set S∗ a = t∗ a · A∗ is a loop transversal in LM(L) to

K;

3 ∀a ∈ E the set S∗∗ a

= {(ta · tx, id)}a,x∈E is a loop transversal in LM(L) to K;

4 ∀a ∈ E the set Sa = {ta · tx}a,x∈E is a loop transversal in L to

R0.

SLIDE 26

Properties of the loop transversal A and proof of PPC

Theorem 33

Let the set A = {tx}x∈E be the loop transversal in the loop L to its subloop R0 from the previous Theorems. Then the following statements are true:

1 transversal A is a group transversal in L to R0; 2 transversal A∗ = {(tx, id)}x∈E is a group transversal in LM(L)

to K;

3 ∀r ∈ R0 a set Sr = r∗ · A∗ · r∗−1 is a group transversal in

LM(L) to K;

4 the group transversal operation (A∗, ·) has a sharply transitive

automorphism group.

Theorem 34

The group (A∗, ·) be a primary cyclic group of order n = pm, p is a prime number.

SLIDE 27

References

[Baer1939] Baer R.: Nets and groups.1. Trans Amer. Math. Soc., 46(1939), 110-141. [Bonetti1979] Bonetti F., Lunardon G., Strambach K.: Cappi di permutazionni, Rend. math., 12(1979), No. 3-4, 383-395. [Burdujan1976] Burdujan I.: Some remarks about geometry of quasigroups (Russian), Mat. Issled., Chishinau, ”Shtiintsa”, 39(1976), 40-53. [Hall1962] Hall M.:Group theory,(Russian) IL, Moscow, 1962. [Johnson1981] Johnson K.W.: S-rings over loops, right mapping groups and transversals in permutation groups, Math. Proc.

Camb. Phil. Soc., 89(1981), 433-443.

[Kuznetsov1987] Kuznetsov E.A.: On one class of ternary system (Russian). In issue “Quasigroups”, Kishinau, “Shtiintsa”, 95(1987), 71-85. [Kuznetsov1994.1] Kuznetsov E.A.: Transversals in groups.1.Elementary properties, Quasigroups and related systems, 1(1994), No. 1, 22-42.

SLIDE 28

References

[Kuznetsov1994.2] Kuznetsov E.A.: Sharply k-transitive sets of permutations and loop transversals in Sn, Quasigroups and related systems, 1(1994), 43-50. [Kuznetsov1995.1] Kuznetsov E.A.: About some algebraic systems related with projective planes, Quasigroups and related systems, 2(1995), 6-33. [Kuznetsov1995.2] Kuznetsov E.A.: Sharply 2-transitive permutation groups.1, Quasigroups and related systems, 2(1995), 83-100. [Kuznetsov1999] Kuznetsov E.A.: Transversals in groups.2.Loop transversals in a group by the same subgroup, Quasigroups and related systems, 6(1999), 1-12. [Kuznetsov2001.1] Kuznetsov E.A.: Loop transversals in Sn by Sta,b(Sn) and coordinatizations of projective planes, Bulletin of AS

f RM, Mathematics, 2001, 2(36), 125-135.

SLIDE 29

References

[Kuznetsov2001.2] Kuznetsov E.A.: Transversals in groups.3.Semidirect product of a transversal operation and subgroup, Quasigroups and related systems, 8(2001), 37-44. [Kuznetsov2002] Kuznetsov E. Transversals in groups.4.Derivation

construction. Quasigroups and related systems, 9(2002), p. 67-84.

[Kuznetsov2005] Kuznetsov E. A loop transversal in a sharply 2-transitive permutation loop. Bulletin of the Acad. of Sci. of Moldova, Mathematics, No. 3 (49), 2005, p. 101-114. [Kuznetsov2010.1] Kuznetsov E.A.: Transversals in loops.1.Elementary properties. Quasigroups and related systems,

No. 1 18(2010), 43-58.

[Kuznetsov2010.2] Kuznetsov E.A., Botnari S.V.: Invariant transformations of loop transversals. 1. The case of isomorphism. Bulletin of the Academy of Sciences of Moldova, Mathematics, 2010, No. 1(62), 65-76.

SLIDE 30

References

[Kuznetsov2011.1] Kuznetsov E.A. Transversals in loops.2.Structural theorems. Quasigroups and related systems, 2011, No. 2 (19), p. 279-286. [Kuznetsov2011.2] Kuznetsov E.A. Transversals in loops.3. Loop

transversals. Bulletin of the Academy of Sciences of Moldova,

Matematica, 2011, nr. 3 (67), p. 3-14. [Kuznetsov2012] Kuznetsov E.A., Botnari S.V. Invariant transformations of loop transversals.2. The case of isotopy. Bulletin of the Academy of Sciences of Moldova, Matematica, 2012, nr. 3 (70), p. 72-80. [Kuznetsov2014] Kuznetsov E.A., Botnari S.V. Invariant transformations of loop transversals.1a. The case of

automorphism. Bulletin of the Academy of Sciences of Moldova,

Matematica, 2014, nr. 1 (74), p. 113-116. [Kuznetsov2016] Kuznetsov E.A. General form transversals in

groups. Bulletin of the Academy of Sciences of Moldova,

Matematica, 2016, nr. 2 (81), p. 93-106.

SLIDE 31

References

[Lorimer1973] Lorimer P.: Finite projective planes and sharply 2-transitive subsets of finite groups, Springer Lect. Notes Series of Math., 372(1973), 432-436. [Pflugfelder1991] Pflugfelder H. O.: Quasigroups and Loops. An

Introduction. Sigma Series in Pure Mathematics (Book 7),

Helderman-Verlag, 1991, 160 p. [Sabinin1972] Sabinin L.V.: About geometry of loops (Russian),

Mat. zametki, 12(1972), No. 5, 605-616.

SLIDE 32