Determination of the Number of Non-abelian Isomorphic Types of Certain Finite Groups

Determination of the Number of Non-abelian Isomorphic Types of Certain Finite Groups

Chapter One

AIM AND OBJECTIVES

The aim of this thesis is to determine the number of non-abelian isomorphic types of certain finite groups of higher orders. We hope to achieve the following objectives:

Finding relationship, through series of examples, of the number of non- Abelian Isomorphic types of groups of order n=sp and the congruence relation between the primes s and
Determining the proof for the number of non-Abelian isomorphic types in each congruence relationship and stating their defining
Determine and design a suitable computer program that will help in working out the number relationship between such primes and generating the numbers for the non-Abelian isomorphic
Finding the non-Abelian isomorphic types of groups of order n = spq where s,p and q are distinct primes and determining their defining

CHAPTER TWO

LITERATURE REVIEW

MULTIPLICATION TABLES OF GROUPS OF ORDER 2 TO 10.

In his work, Wavrik J. (2002) developed a JAVA applet that allows experimentation with group multiplication tables. Here we present some of his work for groups of order 6 and 10. It was noted that any group of order 6 and 10 is isomorphic to one of the groups given below and some their tables are outlined in Tables 2.1 and 2.2 below.

C₆, the cyclic group of order 6 Described via the generator a with relation a⁶ = 1:

Elements:

Order 6: a, a⁵ Order 3: a², a⁴ Subgroups:

Order 6: {1, a, a², a³ ,a⁴, a⁵} Order 3: {1, a², a⁴}

Order 2: {1, a³}

Order 1: {1}

S₃, the symmetric group on three elements

Dcribed via generator a, b

with relations a³ = 1, b² = 1, ba = a^-1b: Elements:

Order 3: a, a²

CHAPTER THREE

METHODS AND GENERATION OF NON-ABELIAN ISOMORPHIC TYPE

Groups factorizable into products of two primes s and p and s,p and q respectively were mainly considered. The use of the list of primes listed in Appendix 1 and the use of the conventional ways of determining the non-abelian isomorphic groups of such orders will also be made.

The scheme in Appendix II was developed to determine the numbers of integer t whose powers of s gave a remainder modulo 1 after division by p in each case.

It is written with HTML and PHP and PHP is Hyper Text Preprocessor and hosted at http://www.cenpece.org/modulo/. HTML is used because it was expected to run on a web browser which is the purpose of maximizing resources which are readily available on web browsers and can always be updated. PHP is a programming language which shares similar syntax with C++, C# and other generic languages. PHP runs seamlessly with database applications such as MySQL and Oracle Database.

It can be run on any kind of system with any form of internet connection or connection of an apache server.

The congruence modulo project can be extended to store a couple of values in the database to make it better for future usage.

Actually, when a group of order is n factorizable into two prime sp such that p º 1 (mod s) and through the relation t^s º 1 (mod p), the scheme gives all the possible values of r in the interval 1< t < p. We will, however, not only outline different values of t but will also put up defining relations of such non-Abelian isomorphic types that would be obtained from different values of r.

This was also done for cases where p º k (mod s) for k > 1.

CHAPTER FOUR

RESULTS

Here we put up our examples and findings from the previous chapter.

RESULT 1. Groups of order 2p have only one non-Abelian Isomorphic

PROOF:

Let G = ‹ a › × ‹ b › such that a² = b^p = 1. Then the non-Abelian isomorphic type must have the relation

CHAPTER FIVE

CONCLUSION AND RECOMMENDATIONS

Our work here was organized in the following manner: First we looked at groups of order 2p, where p is a prime. Since every positive prime is congruent to 1 modulo 2, we did not have much difficulty in out lining the nature of the non-abelian groups of such orders. Next, we used our scheme to look at groups of order n = sp in which case we particularly looked at those prime greater 3 and are congruent to 1 modulo 3. We also tried to display their defining relation in most of the cases.

Armed with our scheme, we also sort for and obtained the number of non-abelian isomorphic types of groups of order 5p, 7p, 11p, 13p and so on. We kept the demand that p is congruent to 1 modulo 5, 7, 11, 13, in all the cases.

From the group of order 15 = 3 x 5, we sought to see what would be the fate of groups whose prime factorization were such that none of the factors if congruent to one modulo the other.

For groups of order n = spq, where s, p, and q are distinct primes, we first considered groups order 30, 42, and 70. One readily observes that such groups are of the form 2pq where each of p and q is congruent to 1 modulo 2 but may not be congruent to 1 modulo each order. We later considered when s ≠ 2. The demand here is not restricted to each of the primes being congruent to 1 modulo others.

SUMMARY OF RESULTS

The area of group classification up to isomorphism and determination of isomorphic types of groups of certain orders is as old as group theory itself.

There is no easy way out hence many tend to pursue it through different approaches. In this Thesis we devoted our work to finding the non-abelian isomorphic types of certain groups of order n = sp, spq and found the following

We developed a scheme that determines the numbers that help to forms the non-Abelian isomorphic types of a group can
We gave with examples proofs of the form of the non-abelian isomorphic types of groups of order 2p, 3p, 5p, 7p,…., and 2pq, 5pq, 7pq,…

CONTRIBUTION TO KNOWLEDGE

That the number of the non-abelian isomorphic types of groups of order n = sp increase as the values of s and p
Why groups of order n = sp, where p is not congruent to 1 modulo s, cannot have a non-abelian isomorphic
That groups of order n = spq have non-abelian isomorphic type irrespective of whether the prime factors are congruent to 1 modulo others, that is whether s divides p -1 and q-1.
That the relationship between the prime factors of the order of groups determine to a large extent whether such groups would have non-abelian isomorphic type or

AREAS OF FURTHER RESEARCH

There is room to further look at groups whose orders are factorizable into more that three
The use of those groups whose prime factors s and p such that p is not congruent to 1 modulo
The possibility of the use of isomorphic types to resolve the fundamental relationship between the underlying biochemistry and the structure of erythrocyte and other cells.
To determine the relationship existing between the different values of r andthe prime p in the non-Abelian isomorphic types of groups of order 3p, 5p, 7p and so on.

To determine the non-Abelian isomorphic types of groups of order n = 11pq, 13pq and so on where p, q > 13.

CONCLUSION

Based on our finding so far we showed that the number of non-abelian groups of order n = sp increase as s and p increase for p congruent to 1 modulo s in all the cases. Again, we see that for n = spq, the non-abelian isomorphic types do increase as s, p and q becomes larger due possibly to the congruent relationship among the prime factors.

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