Relations between the degrees of transitive constituents of G1 and the absolutely irreducible constituents of permutation representation G* of the group G

Let be a finite set of arbitrary elements, G be permutation group on Ω, Δ ⊆ Ω, G1 = {1} = Δ1, Δ2,..., Δn are n orbits of G, ni = |Δi|, fi is the degree of different irreducible representation of D1…Dr appearing in G*, V=V(G) be the ring of all the matrices of G, dimV=k. And also if the irreducible constituents of permutation representation G* are all different, then the expression is a rational integer.

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Introduction

In 1943. R. Brauer studied about permutation groups and finid the permutation groups of prime degree and related classes of Groups (See) [4] . In 1928, M.J. Weiss studied the primitive groups which contain substitutions of prime order p and of degree 6p or 7p, and in 1934 he worked on simply transitive groups and obtained the beautiful consequences, (See) [25,26] . In 1906, W. Burnside introduced d researched the a bout transitive groups, of prime degree (See) [2] , and in 1921, he worked about the certain simply-transitive permutation group and obtained a beautiful consequences (See) [3] . From years 1906 to 1936. W.A. Manning studied about primitive groups and finding the primitive groups of classes six, ten, twelve and fifteen. (See) [11][12][13][14][15][16][17][18] . In 1937 J.S. Frame determined the degrees of the irreducible components of simply transitive permutation groups, and in 1941, he obtained the double cosets of a finite groups (See) [5,6] . And also in 1952, he finding the irreducible representation extracted from two permutation groups (See [7] ). G.A. Miller (1897 & 1915), (See) [19,20] , E.T. Parker (1954), (See) [21] , M. Suzuki (1962), (See) [23] , J.G. Thompson (1959), (See) [24] , M.J. Weiss (1928), (See) [25,26] , H. Wielandt (1935 & 1956) (See) [27,28] and H. Zassenhaus (1935), (See) [30] are studied about transitive and primitive groups and their obtaind the beautiful and more consequence. In 1935, N.Itô, studied on primitive permutation groups and in 1962 he obtained transitive simple permeation groups of degree 2p, (See) [9,10] . Now in this paper we will study the relations between the degrees ni of the transitive constituents of G1 and the degrees i f of the absolutely irreducible constituents Di of the permutation representation G* of the group G, and the end we expressed this relations in expression

, and also our prove that the expression q is a rational integer.

Periliminares:

In this chapter we study the elementary properties, lemmas and theorems, whose we will used the next chapter.

Elementary notions and definitions:

Let  be a finite set of arbitrary elements which for natural numbers 1,2,…,n as the points and  subset of  . Then a permutation on  is a one-to-one mapping of  onto itself.

. We define the product pq of two permutations p and q on  by the formula q p pq ) (

 

. trivialy pq is again a permutation on  . With respect to the operation above, all the permutations on  form a group, the symmetric group S G . Let G be a permutation group on  , in short   S G . We say that a set

. Then each G g  induces a permutation on  which we denote by  g . We call the totality of  g 's formed for all G g  the constituent  G of G on  ( for example

, then the constituents  G is called faithful. Every group G on  has the trivial fixed blocks  on  . If it has no others it is called transitive. Otherwise it is called intransitive.

Accordingly, a constituent  G is transitive precisely when  is a minimal fixed block (   

). In this case  is called an orbit or set of transitivity of G. Every permutation g on  can be regarded in the following way as a liner substitution in |  |=n variables. the variables n 1 X ,..., X are taken as points. We form colum vectors

is the well known Kronecker symbol). We call g* the permutation matrix corresponding to g. such a matrix contains exactly one 1 in each row and column and zeros every where else. In addition, every permutation matrix g* is orthogonal, i.e., the transpose * g  of g* is identical with its inverse:

G* we denote the group of all matrices g* with G g  . Obviously G* is isomorphic to G. We call G* the permutation representation of G. [22] ). If a transitive permutation group G is regarded as a matrix group G*, then the matrices which commute with all the matrices of G* form a ring V=V(G). We call V "the centralizer ring corresponding to G". V is a vector space over the complex number field which has the matrices ) B( corresponding to the orbits  of G1 as a linear basis. In particular, the dimension of V coincides with the number k of orbits of G1.

Theorem (See

Proof: See [29] , Theorem 28.4] Let D1,…,Dr be the different irreducible representations appearing in G* where D1 is the identity representation. In the following we always denote by fi the degree of Di (i=1,…,r), and by ei the multiplicity of Di in G*. In particular, we have e1=f1=1 and n f e (Here the summation is over all orbits of Gi) Proof: See ( [29] , proposition 28.2).

Theorem. V is commutative if and only if all the ei=1.

Proof: See ( [29] , Theorem 29. Proof: See ( [29] , Theorem 29.8) 2.8. Theorem: Let  and  be two orbits of G1. then

See ( [29] , Theprem 28.10)

Main result

In this section by used of the theorems and lemmas of section 2, we prove that the following main theorem:

3.1.Main theorem:

Let  be a finite set of arbitrary elements, G be permutation group on

is the degree of different irreducible representation of D1…Dr appearing in G*, V=V(G) be the ring of all the matrices of G, dimV=k. And also if the irreducible constituents of permutation representation G* are all different, then the expression

Before from proof of the main theorem, we first the prove the following theorems. And the end we consequence the main theorem.

Theorem: (A)

If the irreducible constituents of G* are all different, i.e., if all the multiplicities ei=1, then the rational number

(B) If in addition the k numbers ni are all different, then q is a square. (C) If the irreducible constituents of G* all have rational characters, then q is a square. The hypothesis is always fulfilled if the degrees fi are all different. Notice: It is not known whether (B) is true for k>3.

Proof of 3.2.(A).

It suffices to show that q is an algebraic integer. The notation of the preceding section is continued. Let U again be the unitary transformation matrix introduced in §2. From the hypothesis ei=1 it follows that every matrix - 1 BU with

 is the fi by fi identity matrix.

Let wi be the diagonal elements of the matrix

Iz, w i.e., , w z w we therefore have FI.

I Nn  

By taking the determinant we get

The wij, as eigenvalues of the matrix Bj which has integer coefficients, are algebraic integers, and therefore | I | and I are also algebraic integers.

We wish to show that I is divisible by n. by 2.5, j j B  =M where M is the n by n matrix consisting of n 2 ones. M has the eigenvalue n occurring with multiplicity 1 belonging to the eigenvector

are then rational. The wij are therefore rational. Since the ij w were already in the proof of 3.2 (A) shown to be algebraic integers, they are rational integers. ij w ]  is therefore a rational integer. Since n divides ] n , ] -1 is also a rational integer, and q is therefore a square as was asserted.

(c) The hypothesis that all the irreducible characters appearing in  G are rational is fulfilled if the degrees i f of the irreducible constituents of  G are all different. For since G* is rational, with each irreducible representation i D all representations conjugate to it appear in  G . Because all of the i f are different, these coincide with i D , and i X is therefore rational.

We assume in the sequel that V is commutative, i.e., that the irreducible constituents of  G are all different, and draw some conclusions from 3.2.

Theorem 3.3. If the nontrivial irreducible representations appearing in

 G all have the same degree f, then the orbits of 1 G

different from 1 all have the same length, which also equals f. Hence G is either regular and Abelian (f = 1) or (3/2)-fold transitive (f >1).

The geometric mean of the i n is at most equal to the arithmetic mean, thus f n It is not known if in general in general when the i n are all different (i=2, …,k) it follows that the i f are also all different. Should this prove correct, the conjecture 3.2(B) would be true.

In the proof of theorem 3.2 we used the matrix equation FI I Nn   only to compare the determinants. More precise information can be obtained from the elementary divisors. Since the k-th elementary divisor of a product of matrices is known to be divisible by the k-th elementary divisor of each factor, the k-th elementary divisor of Nn is divisible by the k-th elementary divisor of F. In other words: If in addition the irreducible constituents of  G have rational characters, then q is square. The hypothesis is always satisfied if the degrees of the different irreducible constituents of  G are all different.

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