Received 20 May 2016; accepted 14 August 2016; published 17 August 2016
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1. 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
, 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 con- veniently 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
there is an associated operator
, which acts on the vector space V.) In fact,
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
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
in the collection of
. Furthermore on the discovery of the special permutation group
, several other works have been done to show some interesting results and properties of
. Some of these works include [10] a paper that studied the Algebraic 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 Γ1 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 Γ1 non-deranged permutation group while we study some other results as it relates to representations of groups.
2. Notation
In an attempt to simplify this paper, basic concepts and notation as related to the work are defined below.
Definition 2.1:
Γ1-non deranged permutation group
is a special permutation group with a fixed element on the first column from the left.
Definition 2.2:
A permutation of a set X is a bijective function
. It is a quantity or function that carries n indices or variables (where each can run from
). For instance, Let
be a non-empty ordered set such that
. Let
be a subgroup of symmetry group
, such that every
is generated by arbitrary set
for any prime
using the following
(1)
Lemma 2.3:
The order of the group
p, a prime is
(
).
Proof. We recall that Langrange’s theorem says that order of the group is divisible by the order of the subgroup. If
then
![]()
where q is a positive integer. We claim that
is a factor of
hence
is the order of
□
Example 2.3.1:
For p = 5 Equation (1) will generate a Γ1 permutation group ![]()
![]()
![]()
and written
in cycle form,
![]()
Definition 2.4:
A representation of G over
is a homomorphism
from G to
, for some n. The degree of the
is the integer n. Thus if
is a function from G to
, then
is a representation if and only if
for all
: Since a representation is a homomorphism, it follows that for every representation
, we have
1) ![]()
2) ![]()
for all
, where
denotes the
identity matrix.
Definition 2.5:
Let G be a subgroup of
, so that G is a group of permutations of
. Let V be a n-dimensional vector space over
, with bases
for each i with
and each permutation
define
for all i, and all
is called the permutation module for G over F. We call
the natural basis of V.
Definition 2.6:
Two-line notation is a notation used to describe a permutation on a (usually finite) set. For a finite set sup- pose S is a finite set and
is a permutation. The two-line notation for
is a description of
in two aligned rows. The top row lists the elements of S, and the bottom row lists, under each element of S, its image under
.
If
, the two-line notation for
is:
![]()
Definition 2.7:
Consider a finite set S and an ordering of the elements of S, with the elements (in order), given as
. For a permutation
of S, the one-line notation for
is the string
. 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.
3. Representation of Gp
In considering
-non deranged permutation group
, and its representation. Let
be a group and let
be
or
and let
denotes the group of invertible
matrices with entries in
.
A representation of
over
is a homomorphism
from
to
for some p. The degree of
is the prime p. Thus if
is a function from
to
, then
is representation if and only if
for all
. Since a representation is a homomorphism, it follows that for every representation
, we have
and
for all
where
denotes the
identity matrix
3.1. Gp as FGp-Module
We need to introduce the concept of an
module, and show that there is a close connection between the
module and the representation of this
-non deranged permutation group
over
.
Let
be a group and let
be
. suppose that
is representation of
write
, the vector space of all row vectors
with
for all
and
, the matrix product
of the row vector V with the
matrix
is a row vector in V. (since the product of a
matrix with an
matrix is again
matrix).
3.2. Proposition
Let V be a vector space over
and let
be a group then V is an
-module if a multiplication
(
,
) is defined satisfying the following conditions for all
and ![]()
1) ![]()
2) ![]()
3) ![]()
4) ![]()
5) ![]()
Proof:
1) Let
such that
. Then
![]()
which implies that
and
. Therefore ![]()
2) Let
and
and are given as
and
, then
![]()
3) Let
and ![]()
![]()
4) Let
then
![]()
5) Let
and
and given that
then
![]()
□
3.3. Corollary
Let
be a subgroup of
(
; p a prime) so that
is a
-permutation group of
. Let
be a representation
define as
(
where
), for all
and V a p-dimensional vector space over
, with bases
for each
. Then
is a permutation module over F with natural bases
of V.
Example
Let
and let B denote the basis v1, v2, v3, v4, and v5 of V. If
then
![]()
![]()
And if
then
![]()
We have
![]()
and
![]()
3.4. Character of a Representation
Suppose that
is a representation of a finite group G. With each
matrix
, we associate the complex number given by adding all the diagonal entries of the matrix, and call this number
. The function
is called the character of the representation
.
Suppose that V is an CG-module with basis
. Then the character of V is the function
define by
.
Naturally enough, we define the character of a representation
to be the character of the corresponding CG-module
namely
.
Similarly, suppose that V is an
-module with basis β. Then the character of V is the function
define by
![]()
Naturally enough, we can also define the character of our representation
to be the character of the corresponding
-module
namely
![]()
3.5. Theorem
Let
be a
-non deranged permutation group, (p = prime and p ≥ 5), the character
of
is never zero.
![]()
Proof:
To prove that
, then it’s sufficient enough if we can show at least one diagonal ele-
ment of
since ![]()
Suppose that
and recall that
is defined as
![]()
Therefore the character of every
is at least 1
. □
3.6. Corollary
Every
(where
) has a trivial character.
3.7. Theorem
Let
be a permutation group, (p = prime and p ≥ 5) and
, then the character
of
is
![]()
Proof:
From Corollary 3.6 above the first part of the proof is obvious, for the second part.
Let
be a subgroup of
, so that
is a group of permutations of
. Let V be a p-dimen- sional vector space over
, with basis
from 2.1 it implies that
![]()
then for all p ≥ 5, the representation
will be
![]()
Applying the definition 2.2 and corollary 3.5, then taking the summation of the diagonal elements will give p as the character. □
4. Conclusion
This paper has extended the one line permutation pattern of Abor and Ibrahim (2010) to a two-line notation and hence
is a
-non deranged permutation group with a natural identity. The representation of
as a finite permutation group was done and the character of
was computed to be 1 if
otherwise p.