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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