Primitive Permutation Groups For The Two Orbital Equations

Primitive Permutation Groups For The Two Orbital Equations

PRIMITIVE PERMUTATION GROUPS FOR THE TWO ORBITAL EQUATIONS

Mukesh Chandra

Department of Computer Applications(Mathematics)

I.F.T.M. Moradabad (INDIA)

ABSTRACT

In this paper, the orbital and orbit are basic ideas in group theory. Xu Changliang began to investigate the orbital equations in the primitive permutation groups. There are two equations of orbital to be solved. The aim of this paper is to interpret the

solution to the first orbital equation Pn = Pm and the second orbital equation Pm P P , where m 3. Some general

theory of permutation group is needed for the statement and proof of the result in this paper.

Keywords: Permutation groups , orbital , sequence , monoid , invariant , non-trivial and prime numbers.

We first introduce the basic idea under lying a permutation group form a monoid, the binary relation

invariant Neumann, P.M. and Praeger, C.E. [1] discussed the orbital equation P P P in their study of

three star permutation groups. The aim of this paper is to complete the discussion of two equation of

orbital Pn = Pm and Pm P P , where m 3, Neumann, P.M.[3] began the investigation of the

influence of some equations in this monoid.

In this paper, suppose that S is a non-empty set and let G is a transitive subgroup of S. Consider that the group G acting on a set S, the orbitals are the orbits of G in S x S. The suborbits are orbits of Gt for t S; G is transitive.

The relation

P(t) {x

S t x P

(1.1).

This shows that a one to one corresponding subdegree of G on S. The number of suborbits or orbitals is said to be rank of G.

The paired orbital P of the orbital P is defined as

P {(x1, x 2 ) | (x 2 , x 1 ) P

(1.2)

The orbital P is known as the self-paired if P = P .

The subsets P and T defined as;

P T

{(x1, x2 )

S | ( x S

x x P

(1.3)

and

(x1, x2 ) ,

Since if P and T are orbitals, then P T will be a union of orbitals i.e

P T .

Therefore, for three binary relations P, T and Q

(P T) Q

This is associated with orbitals.

P (T Q)

Notice that if P and T are invariant of G then P T will also be invariant of G. The equality operator E

is an identity element and the relation P as being a non-trivial if P E . In this sense the empty relation is an element of zero, such that P relations.

• P , for all binary

Let the group G is transitive on a set S and the universal set U satisfies zero invariant of group G binary relation P.

P U

U U P

for any non-

In this section, Neumann, P..M.[3] focuses on a discussion of the equations

P2 P P, Pi

P, P3

P2, Pi

P and Pi = E .

Changliang, X. [2] proposed the two equations of orbitals Pn = Pm and Pm

3. The other section contains the solution of these two orbital equations.

P P,

where m

Let us consider that S is a non-empty set and the group G is a transitive subgroup of S. Some general theory of the permutation group is needed for the statement and proof of the results in this section. For the equation of orbital Pn = Pm, if G is strongly primitive on S.

By definition, if a strongly primitive permutation group leaves trivial proper transitive binary relation invariant and we get the following statement.

If G is strongly primitive group on S and G has a non- trivial orbital P for which Pn = Pm, where n > m; then Pm is the universal relation on a set S. i.e.

Pm = U or Pn-m = E

Such that |G| = |S| = p and n m (mod. p), where p is a prime number.

To prove our main result we need a discussion of the orbital equation Pn = Pm [2]. Therefore, we restate the theorems for ready reference.

satisfying T T T . Then T is a non-trivial invariant partial-order relation of G on S i.e. T = E or

T is the universal relation U on S.

T T T

(2.1)

Since T is a transitive relation. It has a corresponding equivalence relation E (T T ) , (say). Because the group G is primitive, this must be E or universal relation U.

Therefore, there are some possibilities for T T

, E, U and U/E. If T T

, then T is a

strict partial-order relation on S. If T = E.

T T

E, then T is a non-strict partial-order relation or

If U/E T

, then E

T T

. Since E T and so that E

T T .

Therefore, T T U i.e. T = U.

Theorem 2.2: Let G is a primitive on S and G has a non-trivial orbital P, Such that Pn = Pm , where n > m. Define q = n m and the natural number N so that q (N-1) < m qN. Then PqN is a non-trivial invariant partial order relation of G on S i.e. Pq = E or P m is the universal relation U on S .

Because qN m and PqN = PqN+qs (2.1) In particular case, when s = N, we get from equation (2.1),then

PqN = P2qN (2.2)

Now we define T = PqN (2.3)

Since PqN is a non-trivial invariant partial order relation of G on S i.e. T = E. Hence T is the universal relation U on S.

When T = E, then it is easy to prove that Pq = E.

Therefore, T is the universal relation U on S, because n > qN, we have

Pn = U (2.4)

If P is a orbital of G and Pm = U then Pn = U for any n > m. Hence the equation Pm = Pn, if G is strongly primitive on S, then the result is clear Pm = U or Pn-m = E.

3. SOLUTION OF THE ORBITAL EQUATION

Pm = P P

Neumann, P.M.[3], the orbital equations in the primitive permutation groups, If G is finite and

P2 =

P P ,then G is two-homogeneous but not two transitive. The finiteness of the S turns out

to be significant for the analysis of the equation P2 =

P P . The result for the orbital equation

Pm = P P , where m 3, applies in both the finite and infinite conditions : m is odd and

G S 2 .

This section focuses on a discussion of the second orbital equation Pm = P P , where m 3. Let

G is a primitive on S and let P be a non-trivial orbital. Consider that

P3 = P P

and

GS 2 .

Pm =

P P , where m 3.

Suppose that the points

x1, x2, x3, …….., x2m

2 S, such that x1

xm and ( xi , xi 1 ) P for

i=1,2,3,,2m-3 and ( x2m 2 , x1 ) P. Then P P .

where m 3

Pm =

P P

(3.1)

Consider that P

We define

P , so that

P P .

(x)

P(x) P (x)

(3.2)

and

(x)

P(x) P (x) {x}. (3.3)

The first result, it is easy to show that either (x1, xm )

P or (xm , x1 )

P . Consider that

(x1, xm ) P and (xm , x1 ) P .

We show that the following equation :

(x1 )

(xm )

(3.4)

Given that any value in P(x1), because (x1, xm )

P and

xm .

But (xm , xm 1), (xm !, xm

2 ), (xm

2 , xm 3 ),…….. ……, (x2m 3 , x2m

2 ), (x2m

2 , x1)

and (x1,

) are in P and

the equation (3.1) holds, we obtain that either (

Therefore

, xm )

P or (xm , )

P and so that

(xm ).

(x1 )

(xm )

(3.5)

In the same way for the sequence of the points x1, x2, x3, ..,xm-1, xm , then we obtain

(xm )

(x1)

(3.6)

From equations (3.5) and (3.6), then we get

(x1 ) (xm )

Hence the result (3.4) is proved correct.

Now we define equivalence relation ( ) on S: if and only if

( ) ( ) (3.7)

Since G is primitive and the relation s obviously an invariant equivalence relation of G. the

relation is universal and trivial. Therefore the relation relation is not trivial and so that it must be universal.

(x1 )

(xm ) holds and x1 xm ,then the

Since

(x1 )

(x2 ) and then x1

(x2 ) .

Hence (x1, x2 )

P , this is a contradiction, so that either (x1, xm )

P or (xm , x1 ) P .

The first result is complete.

The second result, it is easy to prove that

P P

U | E .

In the same way exactly the proof

(x1 )

(xm ) , but considering that either (x1, xm )

P or

(xm , x1 )

P ,we obtain the following result :

P(x1) P (x1) x1

P(xm ) P (xm ) xm

(3.8)

i.e.

(x1 )

(xm )

(3.9)

Now, a new relation ~ on S is define by

~ if and only if

( ) ( ) (3.10)

Since G is the primitive and the relation ~ is obviously a invariant equivalence relation of G and

the relation ~ is universal or trivial. Therefore and so that it is universal.

(x1 )

(xm ) , then the relation ~ is not trivial

For any value X, in S, so either ( X , )

P or (

, X )

P and X . So that U | E

P P .

Therefore (P P ) E

and P P

U | E .

Hence the second result is complete.

Suppose that any three point x1, x2 and x3, such that

(x1, x2 )

then

P and (x2 , x3 )

(x1, x3 )

P . Because P

U / E .

P , we have

x1 x3 ,

Since it follows that

(x1 , x3 )

P P ,

therefore

P2 P P .

If P2 = P then Pm = P2 = P, which is not true. If P2 P P then using the mathematical

induction on k, we see that

P P

Pk for all k 2.

In particular, P P Pm 1 then E Pm , which is not true.

If P2= P then P 2 P , so that P4 P .

Therefore, Pm = P , if n=1 (mod.3)

or Pm = P , if n=2 (mod.3)

or E Pm , if n=0 (mod.3)

which is not true.

Theorem3.2: Let G is primitive on S and P be a non-trivial orbital. Consider that

Pm P P

and m 4 .

Let P

P3 and P P ,then there are points x , x ,……… , x S , such that x x ,

(xi , xi 1)

1 2 m 1 1 m

P for i =1, 2, 3,,m and (x1,xm 1) P .

which satisfy (x ji , x ji 1 )

S for i =1,2,3,..,k-1 and (x j1 , x jk )

P . The action expansion is

3

defined to add the two points in the following way, for the edge (x j1 , x j2 ) , because P P .

We obtain

x j11 and x j12 , which satisfy (x j1 , x j11 )

P , (x j11 , x j12 )

P and (x j12 , x j2 ) P .

Substituting the point xj11 and xj12 to the sequence between xj1 and xj2, we obtain a new sequence of the points xj1, xj11,xj12,..,xjk .

Consider that m is even, we deal with the problem by some applications of action expansion. In the case when n is odd, given an edge (y1,ym+2), we obtain a sequence of four points

y1,ym,ym+1,ym+2 which satisfy (y1 ,ym )

P , (ym ,ym 1)

P and (ym 1 ,ym 2 ) P .

Step by step we obtain m+2 points y1, y2, y3, . , ym+2.

Therefore ( yi , yi 1 )

P, i

1,2,3,4,……., m and Pm

P P .

We obtain that either (y1 ,ym 1)

P or (ym 1 ,y1)

P , since (y1 ,ym )

P , we obtain y1

ym .

If (y1 ,y m 1 )

If (ym 1 ,y1)

P then y1, y2,……., ym+1 are the points which satisfy the assertion of the theorem.

P , we obtain a sequence of three points ym+1,y1 and ym+2, step by step we get a

sequence of m+1, when n is odd ,we deal with the problem by some application of the expansion.

Given any two points x1 and xm+1 which satisfy (x1 , x m 1)

P by applying the expansion, we

obtain that the points xm-1 and xm

S which satisfy (x1 , x m-1)

P and (x m , x m 1)

P .It is clear

that x1 xm .

Now, we get a sequence of the four points x1, xm-1, xm and xm+1, since n is odd, step by step we

obtain a sequence m+1 points x1,x2,x3,,xm+1, such that x1

xm , then (xi , xi 1)

P and

(x1 , x m 1)

theorem.

P for i =1, 2, 3, ., m. Hence we get a sequence of m+1 points as required for the

P m P P and m

4 , then n is odd and G 2 S .

P ,we have Pm

P and Pm

P , by the theorem of Neumann [3];

G

Also

p S and m=1(mod. p).

G S p and m= -1(mod.p) for some prime number p. So that p = 2 and n is odd,

P P .

First, we show that

P P3

(3.1)

Consider that

P P3

, since P

Pm , we choose x1,x2,x3,.,xm+1 from S, such

that (xi , xi 1 ) P for i = 1, 2, 3, .,m and (x m 1 , x 1) P .

Therefore P

Pm , we obtain xm+2, xm+3,., x2m which satisfy (x , x )

P for i = m+1, m+2,

i i 1

,2m-1 and (x 2m , x 1 ) P .

Now the points x1, x2, x3,, x2m, make a cycle of 2m points which is degenerate. For the (n+1)

points x2, x3,..,xm+2, because (xi , xi 1 )

P , for i=2, 3,., m+1 and Pm

P P , either

(x2 , xm 2 ) P or (xm 2 , x2 ) P .

If (x2 , xm 2 )

P and the four points xm+1, x1, x2, xm+2 make it impossible that P P3

. So

that (x2 , xm 2 )

P and therefore (xm 2 , x2 )

P . By the same argument, we obtain the following

results one by one (xm 3 , x3 )

P , (xm 4 , x4 )

P , ., (x2m , xm )

P , and (x1 , xm 1 )

P . Since

(x1 , xm 1 )

P and (xm 1 , x1 )

P are both true .

Hence this contradiction complete the proof and the equation (3.1) is proved correct. Secondly, it is easy to show that

Pm 2

P P

(3.2)

Since P3 is a union of the orbitals, the equation (3.1) gives

P P3 (3.3)

Given a sequence of (m-1) points z1, z2, z3, ., zm-1 of S ,which satisfy (zi,zi+1) for i =1, 2, 3, , m-2 . The action expansion is defined to add the two points in the following way for the edge (z1, z2), because the equation (3.3) holds. There are two points and , which satisfy

(z1 ,

), ( ,

), (

, z2 ) P .

Adding the two points and , we obtain a sequence of m+1 points z 1 , , ,z 2 ,z 3 ,..,z m 1

which satisfy (z1 ,

), ( ,

), (

, z2 ), (z2 , z3 ),…….. ., (zm 2 , zm 1 ) . This means that the equations

and

P m 2

Pm

P m

P P

(3.4)

(3.5)

Since the equation (3.2) is correct.

In the last, when

Pm 2 P

, we have choose 1 ,

2 , 3 ,……… ,

m 1 in S, which

satisfy ( i ,

i 1 )

P for i=1,2,3,,m-2, and ( 1 ,

m 1 ) , since

P Pm (3.6)

We obtain a sequence of the points

1 , 2 ,

3 ,……..,

m 1 which satisfy (

m 1 , 1 )

P and

( i ,

i 1 )

P for i = 1, 2, 3,., m-2 and (

m 1 , 1 ) P .

Therefore, ( 1 ,

m 1 )

P and (

m 1 , 1 )

P , then we obtain

1 1 (3.7)

For the sequence of the points 1 ,

(3.1) and we obtain

2 , 3 ,……… ,

m 1 and

1 , 2 ,

3 ,……..,

m 1 , using the theorem

P P

which is the contradiction.

(3.8)

When

Pm 2

P , from equation (3.2), we obtain

Pm 2 P

(3.9)

Now consider that a sequence of the points w1,w2,w3,., wm+1, which satisfy

w1 wm , (wi , wi 1 )

P and (w1 , wm 1 )

P for i=1,2,3,m. Therefore, (w1 , wm 1 )

P , then

we obtain (wm 1 , w1 ) P .

We have choose wm+2,wm+3,..,w2m-2 ,such that (wi , wi 1 )

m+1,m+2,m+3,..,m-3.

P and (w2m 2 , w1 )

P for i =

So that we obtain a cycle of (2m-2) points w1,w2,w3,.,w2m-2, which may be degenerate.

Therefore, w1 wm , using the theorem (3.1),then we obtain

P P

which is the contradiction.

Corollary 3.1: If G is strongly primitive on S and has a non-trivial orbital P. Consider that

m 3 and

Then m is odd and

Pm

G S .

P P

The method of proof is elementary and purely combinatorial.

Theorem 3.4: Let G is primitive on S and let P be a non-trivial orbital. Consider that

Then G S 2 .

P3 P P

and

P P ,

P3 P

P3 P

By Neumann, P. M.[3], theorem (2.2)

G S p

and 3 1(mod . p) , for some prime number p.

Since if p = 2, then

G S 2 .

When, P

P then we have

P P3

If we choose a sequence of four points x1,x2,x3 and x4 such that (x1,x2), (x2,x3) and (x3,x4) and

also (x1 , x 4 ) P .

Such as we have

P P ,

x1 x3

Using the theorem (3.1) in the case when n = 3 and we obtain

P P

This show that P P is not true.

I would like to thank my supervisor Dr. T. S. Chauhan, Department of Mathematics, Bareilly College Bareilly, for his constant support and nice guidance. Also Mrs. Alka Chandra for being very supportive. The author thanks the referees for comments that helped to improve the presentation of this paper.

1. Neumann, P. M. and Praeger, C. E.: Three- star permutation groups. Illinois J. Math. 47(2003), p.445-452.

2. Changliang, Xu: Two orbital equations in primitive permutation group. Quart. J.Math.57(2006), p.133-138.

3. Neumann, P. M.: Orbital equations in primitive permutation group . Quart. J. Math.57(2006), p.93-103.

4. Dixon, J.D. and Mortimer, B.: Permutation group. Springer, Berlin , 1996.

5. Wielandt, H. : Finite permutation groups. Academic press, New York, 1964.

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