- Email: [email protected]

ON INCIDENCE PROJECTIVE

MATRICES PLANES

OF

FINITE

MONTARON

Bernard

Renix Electronique, Received

227

56 (1985’) 227-237

December

BP 1146, 31036

Toulouse,

Ceder,

France

1984

In Section 1 and 2, we show that any incidence matrix of a finite projective plane (FPP) can be arranged as a special matrix in which many terms are known. The unknown part of this ordered form of an incidence matrix is shown to be formed by permutation sub-matrices having interesting algebraic and combinatoric properties (Section 3). In Section 4, we introduce a special class of FPP called ‘simple’ FPP. The existence of a simple FPP of order n is related to the existence of a particular n-group called ‘projector’ group. Finally, we make a search of projector groups of orders up to 12, and we give examples of simple FPP, with orders up to 8.

1. Introduction The following

properties

characterize

the incidence

matrix

of a F’PP of order

n

[II); (1) the

(see

matrix is (n*+ n + 1, n2+ IZ+ 1) and their terms are zero and one; (2) each row and each column contains exactly n + 1 non-zero terms; (3) for each pair of columns (resp. rows) (jI, j2), there exists a unique row (resp. column) i, such that terms (i, jJ and (i j2) are ones; (4) two incidence matrices which differ only by permutations of the rows or the columns correspond to the same Fl’P.

2. Ordered form of an FPP

incidence

matrix

According to property (4), we may order any FPP incidence matrix, by permuting rows or columns. An ordering process is described in this section, in which the ordered form is obtained after 5 permutation steps. The permutations used at each step are chosen so that they do not modify parts of the matrix which have been ordered at the previous steps. Step 1. By permuting columns, the to II + 1. By permuting rows (except of column 1 are placed on rows 1 to Fig. 1. The sub-matrices Aii are (n, Aoo = 0, 0012-365X/85/$3.30

0

n + 1 ones of row 1 are placed on columns 1 row 1, which is not moved), then n + 1 ones r~+ 1. The matrix has now the form shown in n). Property (3) implies that

(zero matrix). 1985, Elsevier

Science

Publishers

B.V.

(North-Holland)

8. Montaron

228

1

11...11

oo...oo

oo.........oo

oo...oo

J_

’.

A00

A01

. ........

A

.

*10

A11

.........

*1n

0 0 . . . .

. . . .

.

0

.

.

0

.

.

A

Anl

on

i 0 0 .

0 .

. .

. .

.

. .

. .

.

.

0

’. .

no.

A

.*.*....I

ml

0

Fig. 1

Step 2. n ones of row 2 are distributed amongst matrices Aor, A,,*, . . _, Ao,. By permuting columns >n + 1, these n ones are placed in Aor, which first row is now ‘all 1’. By .property (3) the remaining part of A,,r is ‘all 0’. The same process is then applied to A,,, . . _ , A,,. Similarly, by permuting rows >n + 1, one can arrange the ones of columns 2 to n + 1 in such a way that matrices AkO, 1s k s n, have their column k ‘all 1’. This does not modify matrices A,r, . . . , Ao,. So, we get V k 1 c k s n,

Ak,, = ‘AOk, and A,,

is ‘all 0’ except row k which is ‘all 1’.

Taking into account the new form of AOk and A,,,, k # 0, and properties (2), (3), the Aij matrices, i, jf 0, appear now to have exactly one ‘1’ per row and per column: Vi#O,

VjfO,

AijEP,,

where P,, is the set of all (n, n) permutation

matrices.

Step 3. Any permutation of the columns containing the submatrix Arj, for a given jf 0, does not modify the ordered form obtained at Steps 1 and 2. Since Arj is a permutation matrix, we can arrange their columns so that AIj becomes the identity matrix I,. This process is applied independently for all j# 0. Permutations on the rows containing Air, i 2 2, are used to transform these matrices into I,. So, we get Vk, k#O,

Akl=Alk

= I,.

By property

(3), main

Vi, Vj, i, j 2 2, where

On incidence

matrices

diagonals

of matrices

tr(A,)

of finite projective planes

229

A+ i, j > 2, are now ‘all 0’

= 0 or equivalently

A, E I?:,

PE is the set of (n, n) permutation matrices without fixed point. Two (n, m) matrices

Definition.

be antipodal (denoted Lemma

1.

A, B which terms

are zero and one, are said to

A () B) iff no ‘1’ has the same location

in A and in B.

If A, B, C E P,,, then

A()B+AC()BC

and CA()CB.

This lemma will be used in Section 3. By property (3), after Step 3, for all i, i 32 (resp. Vj, j 3 2) the n - 1 matrices Aii, j 2 2 (resp. i 2 2) are pairwise antipodal. Since Aii have an ‘all 0’ diagonal, we get:

Theorem

1. The Aii sub-matrices of the ordered form of the FPP incidence matrix,

are such that: Vi, i 2 2

i

Aij = J, - I,,,

j=2

Vj, j Z- 2

2 Aii = J,, - I,,, i=2

where J,, is the ‘all 1’ (n, n) matrix. Step 4. Let p, q E P,. Then qp is obtained by permuting columns of q, and tpq is obtained by permuting rows of q. p and ‘p are used to permute rows and columns of the incidence matrix as shown in Fig. 2. This transformation does not modify

P

............. .............

Aon

,............

*1Tl

....... ...... . .

A2* .

.

.

.

. A nn

Fig. 2

B. Montaron

230

the properties

of the ordered

vi, iz

1,

ViiSl,

form after

Step 1, 2, 3, since

A,jp = A,j, ‘PA~~P = A,j; ‘pAi

= Aio, ‘pAi 1p = Ai1

and Vi, j>2, Let $ denote

tpAijp E P”,. an equivalence

binary

relation

over Pz, defined

as follows:

Vq,q’EP0,,q$q’tt3pEPJq’=tpqp. We have now proved

the following

theorem:

Theorem 2. After Step 3, one of the Aii matrices for i,j 2 2 (for instance be chosen amongst the leaders of equivalence classes of P2$. This is step 4-the

number

of possibilities

for AZ2 is now quite

A,J

may

small.

Lemma 2. Let q E P”,, then (1) q symmetrical (‘q = q) * Vp E P,, tp q p symmetrical, (2) q antisymmetrical (tq ( > q) + Vp E P,, tp q p antisymmetrical, (3) VP E pm tP 4 P E p”,. If n is odd, there are no symmetrical matrices in P”,. If n is ‘even, the whole set of symmetrical matrices of P”,, forms an equivalence class of PQ$, whose leader is shown in Fig. 3. The following results on leaders are easily deduced from the decomposition of permutation by disjoined antisymmetrical matrices of equation

cycles. The number of equivalence of Pf is equal to the number

where k > 1 and 3

Fig. 3

classes in the set of of integral solutions

(x,, x2, . . . , xk) is related on the main

0

1

1

0

diagonal

to a of L,.

231

On incidence matrices of finite projective planes

”

10

=

3+3+4 3+7 5+5 4+6

:

5 classes

Fig. 4

These sub-matrices are respectively permutation matrices. An example

(x1, x1), (x,, x,), . . . , (xk, xk) and are circular is shown for L,, in Fig. 4.

The number of equivalence classes in the set of Pz matrices which are not symmetrical and not antisymmetrical is equal to the number of integral solutions of equations

where kal, x1=2, x,#2 and ~cx,=zx,~..*~x,. xk) is related to a leader Each vector (x1,. . . , previously. See example in Fig. 5. The number given in Fig. 6 for n G 12.

II=

10:

Example:

Llo =

2+2+2+4 2+2+3+3 2+2+6 2+3+5 2+4+4 2+8

matrix

of choices

6 classes

10=2+2+2+/l r 01100000000 1000000000 0001000000 0010000000 0000010000 0000100000 %0000000100

Fig. 5

L, of P3$

1

as explained

for AZ2 at Step 4 is

232

B. Montaron

S.

n

21 30 4 50 61 70 81 90 10 11 12

1

1 0 1

Fig. 6. S = symmetrical

A.S.

Others

0 1 1 1 7 2 3 4 5 6 9

0 0 c1 1 1 3 3 4 6 Ii 11

leaders,

Total

1 1 2 2 4 4 7 8 12 14 21

AS. = antisymmetrical

leaders.

1

11

00

00

1

111

000

000

000

1 1

00 00

11 00

00 11

0 0

10 10

10 01

10 01

1 1 1

000 000 000

111 000 000

000 111 000

000 000 111

D 0

01 01

10 01

01 10

0 0 0

100 100 100

100 010 001

100 010 001

100 010 001

0 0 0

010 010 010

100 010 001

010 001 100

001 100 010

0 0 0

001 001 001

100 010 001

001 100 010

010 001 100

n=2

n=3

100 00010

01000 00100 00010 00001 10000

01000

1000 00100

00001 10000 01000 00100 00010

00100 00010 00001 10000 01000

OOOOl 10000 01000 00100 00010

01000 00100 00010 00001 10000

00010 00001 10000 01000 00100

00010 00001 10000 01000 00100

01000 00100 00010 00001 10000

00001 10000 01000 00100 00010

00100 00010 00001 10000 01000

00001 10000 01000 00100 00010

00010 00001 10000 01000 00100

00100 00010 00001 10000 01000

01000 00100 00010 00001 10000

00

0000

forms.

10

00001

1

10000

10000

n=

Fig. 7. Ordered

000

0

1000 0001 0010

0010 0001 1000 0100

0001 0010 0100 1000

0010 0001 1000 0100

0001 0010 0100 1000

0100 lOD0 0001 0010

0001 0010 0100 1000

0100 1000 0001 0010

0010 0001 1000 0100

0100

n=4

5

For n = 4 and 5, only A,, sub-matrices

with i, j 2 2 are shown.

of finiteprojectiueplanes

On incidence matrices

Step 5. The leaders The sequence order)

233

at Step 4 for AZ2 have the first row = (0, 1, 0, . . . ,0).

given

of each first row of matrices

A,,,

. . . , Azn forms

(in an arbitrary

the vectors

(0, 0, 1, 0, . . . , 0); (0, 0, 0, 1, 0, . . . , 0); . * * ; (0, . . . ) 0, l), where

the ‘1’ takes

every

place except

columns

1 and 2.

By permuting blocks of n columns of the incidence matrix, it is possible to arrange the matrices A,,, . . . , A*,,, in a different order without modifying the properties of the ordered form obtained after Step 4. We may choose an arrangement so that first row of AZ3 is (0, 0, 1, 0, . . . , 0), etc. and first row of A,, is (0, 0, . . . , 0, 1). Similarly, we can arrange the matrices A32, _ . . , A,,z in a new order. Finally at Step 5: Vi 2 2, first row of Azi and Ai, has a ‘1’ in column

i.

The matrices obtained after Step 5 are the ordered forms of FPP incidence matrix. For small values of n (2,3,4,5) the ordered forms (Fig. 7) exist and are unique. This shows the existence and unicity of FPP of orders 2, 3, 4, 5.

3. Some other properties

of sub-matrices

Aij

Theorem 1 gives relations between matrices A,, i, j 2 2. Since for permutation matrices p, one has p -’ = ‘p, other relations can be established. Lemma 3. Let Airi, A,, A+ Aipj, be sub-matrices of the ordered form of an FPP incidence matrix, with i, i’, j, j’ 2 2 and i # i’; j # j’. Then property (3) applied to the matrix

is equivalent

to

Ai,jtAijAij’tAi,j, This lemma Theorem

( ) I,.

is easily proved

by calculating

tr(Ai,itAijAij,tAi,j,).

2. vi, j 3 2(i # j)

2

AiktAjk = J,, - I,,,

k=2

Vi, j32(i#

j)

f

Aki’Ak; =Jn-&,.

k=2

Proof

follows

directly

from Lemma

1 and Lemma

3.

B. Montaron

234

Let Vo, Vl, . . . , V;-’

be the column vectors defined as follows:

where Ai+l,j+l are sub-matrices of the ordered form of an FPP incidence matrix. For all i, 1 c i =SII - 1, let I+ = [ Vo, Vt, . _ . , Vy-‘I, Theorem square

3. The (n, n) matrices L1, II,, . . . , I+-,

then: form a complete set of latin

of order n (see [l]).

4. “Simple”

FPP

Definition. An FPP is called simple if the Aii sub-matrices of the ordered form of the incidence matrix have the following properties: (1) Sub-matrices Aij, i, j 5 2 form a latin square; each row Aiz, . , . , A, and each column Azi, . . . , A, is formed of the same set E of n - 1 matrices; (2) The set {I,,}UE is a multiplicative group. As shown in Fig. 7, FPP of orders 2, 3, 4, 5 are all simple. Definition. Let G be a multiplicative group. G is called a projector group iff there exists a latin square T = (tij) of order n - 1, rows and columns of which are formed of set G -{l} such that ti&tij,t;;

# 1,

Vi, i’, j, j’, if

i’, j# j’.

Latin square T is called projector. Since T can be ordered by permutation

of rows and columns, the first row and the

first column may be ordered arbitrarily. Definition. The projector index p of an n-group G is the maximum number of complete rows of a projector table T, that it is possible to construct. p is such that lCpq:n-1. Theorem projector

4. There n-group,

exists a simple FPP of order n ijf there exists a multiplicative formed by (n, n) permutation matrices which sum is J,.

Proof follows directly from definitions (simple FPP and projector group), and from Theorem 1, Lemma 3. The projector latin square T of the group of

On incidence

Theorem

matrices

of finite projectiue planes

4 forms the latin square of A, I,’ J‘>-. 2 sub-matrices

235

of the ordered form

of the incidence matrix.

4. Projector

groups of order s12

All group types of order ~12 are known (see [2]). For each type of group shown in Fig. 8 we tried to construct projector tables T, with as many rows as possible. No simple FPP of order 6 or 10 exists, since there are no projector

groups of

order 6 or 10. GROUP TYPE

Fig. 8. Projector

CROUP PROJECTOR II'!DEX "

CROUP TYPE

GROUP PROJECTOR ORDER I'dEX n P

(See I;‘])

ORZER

Z/2Z (.) Z/32 (.) z/42 2/2ZxZ/?Z(.) Z/57 (.) Z/62. D3 = 53 Z/77. c.1 Z/riZ Z/42x2/27.

2 3 4 4 5 6 6 7 8 F!

1 2 1 3 4 1 1 6 1 3

w7Z)3

8

7

D6

c12.

(+) = to be proved;

(.)

index of group of order

2

(See 121) 02 D4 Z/w, Z/3ZXZ/3Z(.) Z/lOZ Z/llZ (.) Z/l?? Z/2ZXZ/6Z A4 03

2

3

4

4

6

13

5

R n 9 9 10 11 12 12 I2 12

2 7 2 8 1 10 1 a4 2(+) l(+)

12

2(+)

(.) = projector

groups

(p = n - 1).

6 5

362514 4

15

5

316

2

6

3

4

2

Fig. 9

5. Simple FPP of order 7 There exists a unique simple FPP of order 7, based on a multiplicative projector group isomorphic to Z/72. The projector latin square of Z/72 is shown in Fig. 9. Let Mi be a permutation matrix (7,7), corresponding to the element i of Z/72. M,, = I,, M, is the circular permutation matrix and Mk = M’; for k = 2, . . . ,6. The multiplicative projector 7-group of Theorem 4 is G = {MO, Ml, . . . , Me}. One has M,+M,+. * . + M6 = J,. The sub-matrices A,, i, i 3 2 of the ordered form of the FPP incidence matrix are obtained by replacing i by Mi in the projector latin square of Fig. 9.

236

B. Montaron

6. Simple FPP of order 8

Let:

E=I,,

Let:

MO = 112,

A=

M’,[;

,“I,

M,=[;

then, we get M,,+M,=***+M,=J, projector g-group of permutation

and G = {MO, . . . , M,} is a multiplicative matrices, isomorphic to (Z/2Z)3. The projector latin square of G is obtained by replacing i by Mi in the projector latin square of (Z/22)’ shown in Fig. 10. Exactly 8 different projector latin squares of (Z/22)” can be built. I.

2

3

4

2

6

4

13

7

5

6

12

3

7

7

6

7

7

n

1

4

5

3

3

4

4

15

5

3

6

6

7

17

7

5

2

613

5

6

7

7

5

S

Fig. 10

7. Simple FPP of orders 9 and 11 There

exist many simple

FPP

of order

9 based on a multiplicative

group

isomorphic to the projector group (Z/32)‘. There exists a unique simple FPP of order 11 based on a multiplicative group isomorphic to Z/llZ. 8. Order 12 The unique candidate for projector 12-group is Z/2ZX Z/62. A multiplicative permutation matrice 12-group isomorphic to Z/2ZX Z/62 is

Fig. 11

On incidence

G = {A& M,, . . . , i&},

matrices

of

finite projectiue planes

231

with

One has Mo+M1+~~~+M,1=J12. Unfortunately, up to now, we built only 4 row-projector tables. One example of these partial projector tables is obtained by replacing i by Mi in Fig. 11. Applying Theorem 7 to this partial ordered form of an hypothetical incidence matrix, we get a set of 4 latin squares of order 12, pairwise orthogonal. It would be interesting to check the existence of simple FPP of order 12 using a computer.

References [ 11 D.R. Hugues and F.C. Piper, Projective Planes [2] A. Bouvier and D. Richard Hermann, Groupes

(Springer, (Herman,

Berlin, 1973). Paris, 1979).

Copyright © 2020 COEK.INFO. All rights reserved.