Advances in Pure Mathematics, 2013, 3, 292-296
http://dx.doi.org/10.4236/apm.2013.32041 Published Online March 2013 (http://www.scirp.org/journal/apm)
Topological Dynamics in Tandem with
Permutation Groups
Isaac Kwame Dontwi, William Obeng-Denteh, Stephen K. Manu, Richard Nyarko Yeboah
Department of Mathematics, College of Science, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
Email: ikedontwi@hotmail.com, obengdentehw@yahoo.com, deoxy2005@yahoo.com, aricky2008@yahoo.com
Received September 14, 2012; revised November 16, 2012; accepted December 3, 2012
ABSTRACT
The purpose of this study was to delve into the aspects of abstract algebra that has a link with topological dynamics in
terms of permutation and symmetric groups. This would aid users to appreciate the role it plays in the theory and appli-
cation of topological dynamics. The usage of matlab programming to carry out the permutations was carried out. The
study contributes to the literature by providing candid explanation and usage of data-based evidence documenting the
extent to which topological dynamics operates.
Keywords: Permutation Groups; Recurrence; Topological Dynamics; Matlab Code; Flows
1. Introduction
Topological dynamics in [1] and [2] form an aspect of
the theory of dynamical systems in [3] where qualitative
[4], asymptotic [5] properties [6-9] of dynamical systems
are taken into account from the viewpoint of general to-
pology in [10]. Reference [2] provides the first self-con-
tained comprehensive exposition of the theory of dyna-
mical systems as a core mathematical discipline closely
intertwined with most of the main areas of mathematics.
Topological dynamics is defined as the study of asymp-
totic [11-14] or long term properties of families of maps
in [15] of topological spaces. Abstract topological dy-
namics is usually developed in the context of flows as in
[16]. A flow
,
X
TT
is a jointly continuous action as in
[17] of the topological group [18] on the topological
space
X
[19]. This means that there is a continuous
map [20] from
,,
X
TX tex xx tx with
and which clearly indicates that e is the
identity of and

,s tsx
,,Tts
tx T
X
. A topological dynami-
cal system is the central point in the study of topological
dynamics. This constitutes a topological space, in tandem
with a continuous transformation, a continuous flow, in
other words a semigroup of continuous transformations
of that space. This state of affairs brings the activity of
permutation groups [21-23] and semigroups action [24]
into the scenario.
Recorded studies of permutations of the earliest peri-
ods can be found in Sefer Yetsirah or Book of Creation
which was written by an unknown Jewish author some-
time before the eighth century [25]. This was improved
upon with the passage of time.
2. Problem Statement and the Way Forward
The main problem that was considered was to identify
the link between topological dynamics and permutation
groups. The usage of permutation groups was used syn-
onymously with Cartesian product. It became necessary
to decipher the link based on the underlying factors enu-
merated stemming from the laid down rules from credi-
ble sources [26]. A periodic point returns to itself every
hour on the hour; but almost periodic point returns to a
neighbourhood every hour within the hour. This is what
has been established to lend credence to topological dy-
namics in tandem with permutation groups.
3. Preliminary Notations and Definitions
A topological transformation group, or transformation
group is explained to be an ordered triple
,,πXT
π
such
that the following conditions are satisfied:
1) X is a topological space which is called the phase
space: T is called the topological group known as the
phase space and is a map of the cartesian product
X
T
X
, called the phase map;
into
2) for allex xX
π
where e denotes the identity
element of T, and the value of at the point
,of
x
tXT
x
be represented by t
π
;
3) is continuous.
In analyzing a transformation group ,,πXT the
phase map
,
x
txt gives the values of two kinds of
C
opyright © 2013 SciRes. APM
I. K. DONTWI ET AL. 293
maps when one of the variables x, t is replaced by a con-
stant. They are transition and motion in [27]. Depending
on additive groups of real numbers or integers a con-
tinuous flow or discrete flow arises. These two kinds of
flows are related in that a continuous flow determines
many discrete flows by taking cyclic subgroups of R.
Conversely, a discrete flow determines a continuous flow
when the phase space X is extended to the Cartesian
product of X and the closed interval.
The crux of the work here cruises on recurrence. Sup-
pose is a homeomorphism of X onto X, a point x of
the phase space is said be almost periodic under
and
is said to be almost periodic at x provided that if U is
a neighbourhood of x, then there exists a relatively dense
subset A of additive group of integers such that
n
x
U
nA
S
D
for all . A periodic point returns to itself every
hour on the hour; but almost periodic point returns to a
neighbourhood every hour within the hour. This is what
has been established to lend credence to topological dy-
namics in tandem with permutation groups.
4. Methods
A permutation group is a group of permutations on a
finite set of positive integers. Permutation groups are
therefore all operations for groups which can be applied
to them. Here permutation multiplications are carried out
on permutations of the main set. 3 is seen as a group
3 of symmetries of an equilateral triangle in which
two copies of it with vertices 1, 2, and 3 can be placed
with one covering the other with vertices on top of verti-
ces in [28]. The subscripted Greek letters i
and i
are for rotations and mirror images in bisectors of angles.
The second one is the 4 which is the group of sym-
metries of the square. The subscripted Greek letters
ii
D
,
and i
are for rotations, mirror images in per-
pendiculars bisectors of sides, and for diagonal flips re-
spectively. Matlab was then used to develop codes for
the permutations.
5. Results and Discussion
Two important examples would be explored to unearth
the surprising revelations in the processes. See Figure 1
for the operations of the matlab code. Considering the
group 3with elements where S3! 6
1, 2,3.A
2
3
123
312
123
213






00 0
01 1
02 2
123123 123
123123 123
123 123123
123 231231
123 123123
123 312312
The permutations are listed with names using Greek let-
ters assigned to them viz.;















3
S
01
12
123 123
,,
123 231
123 123
,,
132 321










Now,
The detailed process of the permutations can be found
in [28].
The multiplication table for is depicted in Table
1.
The next is to form the dihedral-group 4 of permu-
tations to the ways that two copies of a square with ver-
tices 1-4 can be placed with one covering the other with
vertices on top of vertices. is the group of symme-
tries of the square.
D
4
D
01
23
12
12
1234 1234
,,
1234 2341
1234 1234
,
3412 4123
1234 1234
,,
2143 4321
1234 1234
,
3214 1432




 

 
 












00
0
01
1
02
2
1234 1234
1234 1234
1234
1234
1234 1234
1234 2341
1234
2341
1234 1234
1234 3412
1234
3412










 
 
 











S
The following representations are used;
Now
The multiplication table for is shown in Table 2.
4
Theorem: (Cayley’s Theorem) every group is isor-
morphic to a group of permutations. An important addi-
ion is that in group theory, Cayley’s theorem which was t
Copyright © 2013 SciRes. APM
I. K. DONTWI ET AL.
Copyright © 2013 SciRes. APM
294
Figure 1. Pou1 has been depic ted.
Table 1. The multiplication table for 3
S
.
0
1
2
1
2
3
0
0
1
2
1
2
3
1
1
2
0
3
1
2
2
2
0
1
2
3
1
1
1
2
3
0
1
2
2
2
3
1
2
0
1
3
3
1
2
1
2
0
S
Table 2. The multiplication table for 4.
0
1
2
3
1
2
1
2
0
0
1
2
3
1
2
1
2
1
1
2
3
0
1
2
2
1
2
2
3
0
1
2
1
2
1
3
3
0
1
2
2
1
1
2
1
1
2
2
1
0
2
3
1
2
2
1
1
2
2
0
1
3
1
1
1
2
2
1
3
0
2
2
2
2
1
1
3
1
2
0
I. K. DONTWI ET AL. 295
named in honour of Arthur Cayley, states that every
group G is isomorphic to a subgroup of the symmetric
group acting on G in [28]. In fact this can be explained as
an instance of the group action of G on the elements of G.
A permutation of a set G is any objective function taking
G onto G; and the set of all such functions forms a group
under function composition, called the symmetric group
on G, and this denoted by Sym(G).
Cayley’s theorem outlines all groups on the same ped-
estal, by taking into account any group which might be
made up of infinite groups such as (R,+) as a permutation
group of some underlying set. This establishes the fact
that theorems which holds for permutation groups are
counted to hold for groups in general.
The group being used here satisfies the axioms for
topological group. In topological dynamics it involves
topological groups and their operations [2,27].
6. Using Matlab to Compute the Results
above Instead of Manual Approach
Algorithm
In general this work has been trying to establish a link
between topological dynamics and permutation groups
which has been likened to Cartesian product. Matlab has
been used to generate the permutations explained above.
See Figure 1 for an example of the usage of the matlab
code. The code could be extended to cover Sn.
7. Conclusions
The crux of the work here cruised on recurrence in [29].
A periodic point returns to itself every hour on the hour;
but almost periodic point returns to a neighbourhood
every hour within the hour. The recurrence was depicted
in the permutations that were done. All the parameters in
the form of the Greek symbols recurred in the table pro-
vided. When this trend is continued for all time the re-
currence nature would be mimicked along the same trend.
This is what has been established to lend credence to
topological dynamics in tandem with permutation groups.
It is worthy to note that from all the tables each col-
umn gives a permutation of the group set. It is obvious
that at least every finite group is isomorphic to a sub-
group of the group. Isomorphism is about one-to-one
correspondence satisfying a particular relation. This is
the way topological dynamics apply permutation in find-
ing the symmetries of objects and their rotations as well
and matlab has been useful over here.
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