On the Representations of Γ 1-Nonderanged Permutation Group p

Representation theory is concerned with the ways of explaining or visualizing a group as a group of matrices. In this paper, we extend the permutation pattern of ( )( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 1 i mp mp mp i i p i ω = + + + −  to a two-line notation. We consider the representations of this Γ1 non-deranged permutation group p  (p ≥ 5 and p a prime). Also we reveal some interesting properties and results of the character ( ) i χ ω of p  where i p ω ∈ .


Introduction
The beauty of Group as a topic is the various properties that can arise from its studies.Its interesting nature has encouraged various studies in this field over the years.For instance, for every n a positive integer, the set of all permutations of { } 1, 2, , n  , under the product operation of composition is a group.This group is known as a symmetric group (Permutation group) of degree n.According to [1], the study of the symmetric group by Georg Frobenius in 1903, opened the door to the various works that was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Schtzenberger and Richard P.
Conscious efforts by different researchers over the years led to the discovery of other form of permutation patterns, groups and their subsequent representations; [2] shows how functions acting on a finite set can be conveniently expressed using matrices, whereby the composition of functions corresponds to multiplication of the matrices.Essentially, they considered the induced action on the vector space with the elements of the set acting as a basis.This action extends to tensor powers of the vector space and can be extended also to symmetric powers, antisymmetric powers, etc., that yielded representations of the multiplicative semi-group of functions and representations of permutation groups.
To be precise, [3] described a representation as a homomorphism from G into a group of invertible matrices.[4] described a representation as an (linear) action of a group or Lie algebra on a vector space.(Say, for every g G ∈ there is an associated operator ( ) , which acts on the vector space V.) In fact, π is a representation of G acting on the space V.Most of the informations contained in the representation of a group can be distilled into one simple statistic, the trace of the corresponding matrices; [5].
Over the years, deranged permutation, a permutation with no fix points has been studied with various results established.One of the many works in this field is the group theoretical interpretation of Bara'at Model by [6] to establish a deranged permutation pattern.The theoretic and topological properties have also been studied and established by [7].More recently is the use of Catalan numbers by [8] to develop the scheme for prime numbers 5 P ≥ and which generate the cycles of permutation patterns using to determine the arrangements.This permutation pattern was further worked upon by [9] to establish a permutation group.This was achieved by embedding an identity element { } . Furthermore on the discovery of the special permutation group p G , several other works have been done to show some interesting results and properties of p  .Some of these works include [10] a paper that studied the Alge- braic properties of the (132)-avoiding class of this permutation pattern and its applications to graph.The comparison of the group permutation pattern and generalized permutation patterns using Wilf-equivalence has also been studied by [11].
Besides, as established by [12], that not every transitive group contains a derangement.Hence we will in this paper, take a lead from the representations of symmetry groups as shown by [13] [14] and [15] to show the representations of Γ 1 non-deranged permutation group; this will be achieved by extending the work of [8], to a two-line notation; we will also introduce another identity element for this Γ 1 non-deranged permutation group while we study some other results as it relates to representations of groups.

Notation
In an attempt to simplify this paper, basic concepts and notation as related to the work are defined below.
Definition 2.1: Γ 1 -non deranged permutation group p  is a special permutation group with a fixed element on the first col- umn from the left.

Definition 2.2:
A permutation of a set X is a bijective function : X X ρ → .It is a quantity or function that carries n indices or variables (where each can run from 1, , N  ).For instance, Let Ω be a non-empty ordered set such that be a subgroup of symmetry group n S , such that every i ω is generated by arbitrary set Ω for any prime 5 p ≥ using the following ( ) ( ) ( ) ( )

Lemma 2.3:
The order of the group p ). Proof.We recall that Langrange's theorem says that order of the group is divisible by the order of the subgroup.
where q is a positive integer.We claim that 1 p − is a factor of !p hence 1 p − is the order of .
Since a representation is a homomorphism, it follows that for every representation ( ) , , , n S a a a =  , the two-line notation for ρ is: ( ) ( ) ( ) ( ) Consider a finite set S and an ordering of the elements of S, with the elements (in order), given as 1 2 , , , n a a a  .
For a permutation ρ of S, the one-line notation for ρ is the string ( ) i a ρ . The one-line notation for a permutation is a compressed form for the two-line notation where the first line is omitted because it is implicitly understood.

Representation of p
In  is representation of p  write p V =  , the vector space of all row vectors ( ) λ ∈ for all v V ∈ and i p ω ∈ , the matrix product ( ) i v ω ρ of the row vector V with the p p × matrix i ω ρ is a row vector in V. (since the product of a 1 p × matrix with an p p × matrix is again 1 p × matrix).

Proposition
Let V be a vector space over  and let p  be a group then V is an , .

Character of a Representation
Suppose that ( ) is a representation of a finite group G.With each n n × matrix ( ) we associate the complex number given by adding all the diagonal entries of the matrix, and call this number

Theorem
Let p  be a 1 Γ -non deranged permutation group, (p = prime and p ≥ 5), the character To prove that ( ) the leading diagonal element for 1 Therefore the character of every [ ] i ρω is at least 1 ( )

Corollary
Every i p ω ∈ (where 1 i ≠ ) has a trivial

Theorem
Let p  be a permutation group, (p = prime and p ≥ 5) and 1 e ω = , then the character χ of i p ω ∈ is ( ) Applying the definition 2.2 and corollary 3.5, then taking the summation of the diagonal elements will give p as the character.□

Conclusion
This paper has extended the one line permutation pattern of Abor and Ibrahim (2010) to a two-line notation and hence p  is a 1 Γ -non deranged permutation group with a natural identity.The representation of p  as a finite permutation group was done and the character of p  was computed to be 1 if i e ω = otherwise p.
-line notation.We consider the representations of this Γ1 non-deranged permutation group p  (p ≥ 5 and p a prime).Also we reveal some inter- esting properties and results of the character ( )

□
Example 2.3.1:Forp = 5 Equation (1) will generate a Γ 1 permutation group 5  character of the representation ρ .Suppose that V is an CG-module with basis β .Then the character of V is the function : that V is an p  -module with basis β.Then the character of V is the function :

.
considering 1 Γ -non deranged permutation group p  , and its representation.Let p  be a group and let  be  or  and let  for some p.The degree of ρ is the prime p.Thus if ρ is a function from p p module, and show that there is a close connection between the p  module and the representation of this 1 Γ -non deranged permutation group p  over  .Let p  be a group and let  be  .suppose that ( )  and V a p-dimensional vector space over  , with bases 1 and let B denote the basis v 1 , v 2 , v 3 , v 4 , and v 5 of V.If above the first part of the proof is obvious, for the second part.Let p  be a subgroup of n S , so that p  is a group of permutations of { }