## ABSTRACT

The first part of this work established, with examples, the fact that there are more than one non-abelian isomorphic types of groups of order n = sp, (s,p) = 1, where s<p and

p 1 (mod s) for 100 < p < 4000. The factors s and p are distinct primes. Specifically considered here are groups of order n = 2p, 3p, 5p, 7p, 11p and 13p. It was discovered that the number of non-abelian isomorphic types of groups of order n = sp, s<p increased as n increased. The defining relations of such non-abelian isomorphic groups were outlined and a scheme developed to generate the numbers for the non-abelian isomorphic types of such groups. The scheme helped in generating many examples of non-abelian isomorphic types of such groups. The situation where p k (mod s), k > 1 was worked out and such groups have no non-abelian isomorphic types. This gave credence to the fact that a group of order 15 and its like do not have a non-abelian isomorphic type. It also generated the non-abelian isomorphic types of groups of order n = spq, where s, p and q are distinct primes considering the congruence relationships between the primes. It was seen that there are more non-abelian isomorphic types when q 1 (mod p), q 1 (mod s) and p 1 (mod s). When q is not congruent to 1 modulo p but congruent to 1 modulo s fewer non-abelian isomorphic types were obtained. Moreover, if q is not congruent to 1 modulo p, q not congruent to 1 modulo s, and p not congruent to 1 modulo s, there cannot be a non-abelian isomorphic type of a group of order n = spq. In this case groups of order

n = 2pq, 3pq, 5pq and 7pq were considered. Later, proofs of the number of non-abelian isomorphic types for n =sp and n =spq using the examples earlier generated were given.

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## TABLE OF CONTENTS

TITLE i

DECLARATION ii

CERTIFICATION iii

ACKNOWLEDGEMENT iv

DEDICATION v

TABLE OF CONTENTS vi

LIST OF NOTATION xi

LIST OF TABLES xii

ABSTRACT xiii

CHAPTER ONE

INTRODUCTION

1.0 BACKGROUND OF STUDY 1

1.1 DEFINITION OF PROBLEM 1

1.2 AIM AND OBJECTIVES 3

1.3 SCOPE OF THE STUDY 3

1.4 DEFINITION OF BASIC CONCEPT OF ISOMORPHIC GROUPS 3

1.4.1 Examples 5

1.4.2 Properties of Isomorphic Groups 6

1.4.3 Theorem (Lagrange’s Theorem) 7

1.4.4 Theorem (Cauchy’s Theorem) 7

1.4.5 Theorem (Sylow’s first theorem) 7

1.4.6 Definition 1 7

1.4.7 Definition 2 7

1.4.8 Definition 3 7

1.4.9 Definition 4 7

1.4.10 Theorem (Sylow’s second theorem) 8

1.4.11 Theorem (Sylow’s third theorem) 8

1.4.12 Theorem (A Basis Theorem for Finite Abelian Groups). 8

1.4.13 Theorem (Another Basis Theorem for Finite Abelian Groups) 8

1.4.14 Theorem 8

1.14.15 Proposition 9

1.4.16 Theorem (Frobenius) 10

1.4.17 Remark 10

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1.4.18 Polynomial 10

1.4.19 Theorem 11

CHAPTER TWO

LITERATURE REVIEW

2.0 MULTIPLICATION TABLES OF GROUPS OF ORDER 2 TO 10. 12

2.1 ISOMORPHIC TYPES OF SMALL GROUPS OF ORDER n=sp 15

2.2 PROPOSITION 17

2.3 COROLLARY 18

2.4 GROUPS OF ORDER p2q 18

2.4.1 Proposition 21

2.4.2 Summary of Defining Relations 21

2.4.3 Remark 22

2.4.4 Proposition 23

2.4.5 Summary of Defining Relations 24

2.4.6 Proposition 25

2.4.7 Summary of Defining Relations 25

2.4.8 Proposition 27

2.4.9 Summary of Defining Relations 28

CHAPTER THREE

METHODS AND GENERATON OF NON-ABELIAN ISOMORPHIC TYPES

3.0 NON ABELIAN TYPES OF SOME GROUPS OF ORDER 2p. 32

3.1 NON-ABELIAN GROUPS OF ORDER n=3p WITH

100 < p < 2000 and p 1(mod 3) 34

3.2 LEMMA 41

3.3 FOR SUBGROUPS OF ORDER 3p WHERE 2000 < p < 4000 41

3.4 LEMMA 43

3.5 GROUPS OF ORDER 5p WHERE p 1(MOD 5)

WHERE 100 < p < 2000. 43

3.6 LEMMA 47

3.7 FOR GROUPS OF ORDER n = 5p, FOR 2000 < p < 4000. 47

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3.8 LEMMA 49

3.9 FOR GROUPS OF ORDER n = 7p SUCH THAT p 1 (MOD 7)

AND 20 < p < 2000. 50

3.10 LEMMA 55

3.11 FOR SUBGROUPS OF ORDER 11p 55

3. 12 LEMMA 56

3.13 FOR SUBGROUPS OF ORDER 13p, WHERE p 1 (MOD 13) 56

3.14 LEMMA 57

3.15 GROUPS OF ORDER n = sp WITH NO NON-ABELIAN

ISOMORPHIC TYPES. 57

3.16 LEMMA 62

3.17 THE NON ABELIAN ISOMORPHIC TYPES OF GROUPS OF

ORDER n=spq 62

3.18 SUMMARY OF DEFINING RELATIONS 66

3.19 PROPOSITION 67

3.20 LEMMA 69

3.21 FOR THE GROUPS OF ORDER n=3x (7q) 71

3.22 LEMMA 74

3.23 GROUPS OF ORDER 3x5xq 75

3.24 LEMMA 75

3.25 LEMMA 76

3.26 REMARK 76

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CHAPTER FOUR

RESULTS

4.1 RESULT I 77

4.2 RESULT II 78

4.3 RESULT III 78

4.4 MAIN RESULT 79

4.5 MOTIVATION 79

4.6 CORROLARY 82

4.7 RESULT FOR GROUPS OF ORDER n = sp SUCH THAT p IS

NOT CONGRUENT TO 1 MODULO s. 83

4.8 RESULT 83

4.9 REMARK 85

CHAPTER FIVE

CONCLUSION AND RECOMMENDATIONS

5.1 SUMMARY OF RESULTS 86

5.2 CONTRIBUTION TO KNOWLEDGE 87

5.3 AREAS OF FURTHER RESEARCH 87

5.4 CONCLUSION 88

REFERENCES 89

APPENDIX I 90

APPENDIX II 91

## CHAPTER ONE

INTRODUCTION

1.0 BACKGROUND OF STUDY

Group Theory is relevant to every branch of Mathematics where symmetry is studied. Every symmetrical object is associated with a group. It is due to this association that groups arise in different areas like Quantum Mechanics, Crystallography, Biology, and even Computer Science. There is no such easy definition of symmetry among objects without leading its way to the theory of groups. Classifying groups arise when trying to distinguish the number of isomorphic groups of order n. In organic chemistry, conformal factors affect the structure of a molecule and its physical, chemical and biological properties. For instance, the position of atoms, relative to one another affects the structural formula of Hydrogen peroxide, H2O2. We could write two different planar geometries that differ by a 1800 rotation about 0 – 0 bond. According to Francis A Carey (2003) one could also write an infinite number of non planar structures by tiny increments of rotation about the 0 – 0 bonds; Francis A Carey (2003). Groups may be presented in several ways like multiplication table, by its generators and relations, by Cayley graph, as a group of transformations (usually a geometric object), as a subgroup of a permutation group, or a subgroup of a matrix group to mention a few.

1.1 STATEMENT OF THE PROBLEM

Classifying groups arise when trying to distinguish the number of isomorphic types of a group of order n.

Hall Jnr and Senior (1964) used invariants as the number of elements of each order k (k small) to determine whether two groups of order 2n (n < 6) are isomorphic. Philip (1988) in his article developed a systematic classification theory for groups of prime

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power orders. For certain classes of groups, there exists practical methods to list such groups. Newman and O’Brien (1990) introduced an algorithm to determine up to isomorphism the groups of prime-power order. The determination of all groups of a given order up to isomorphism is an old question in group theory. It was introduced by Cayley who constructed the groups of order 4 and 6 in 1854.

MeubÜser (1967) listed all groups of order at most 100 except for 64 and 96. The groups of order 96 were added by Lane (1982).

Moreover, for factorizations of certain orders, the corresponding groups have been classified, e.g. Holder (1983) determined the groups of order pq2 and pqr, and James (1980) determined the groups of order pn for odd primes and n < 6.

Recently, algorithms have been used to determine certain groups. For example O’Brian (1991) determined the 2-groups of order at most 28 and the 3-groups of order at most 36. Moreover, Betten (1996) developed a method to construct finite soluble groups and used his construction to construct soluble groups of order at most 242.

Determination of isomorphic types has been a comparatively difficult problem as there was no method that is sufficiently effective.

Most of the classifications of the non-abelian isomorphic types of certain finite groups were done for groups of small orders. This is possibly due to the complexity of computation as the factors increase. The problem then arise to find the non-abelian isomorphic types of groups of higher orders which can be factorized into two or three distinct primes taking into consideration of the relationship between the prime factors. The need also arise to construct a suitable computer program to assist in solving such a problem.

Hence, the statement of the problem is “Determination of the Number of non-Abelian Isomorphic Types of Certain Finite Groups”.

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1.2 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:

(i) 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 p.

(ii) Determining the proof for the number of non-Abelian isomorphic types in

each congruence relationship and stating their defining relations.

(iii) 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 types.

(iv) Finding the non-Abelian isomorphic types of groups of order n = spq where

s,p and q are distinct primes and determining their defining relations.

1.3 SCOPE OF THE STUDY

The scope here is limited to the determination of the number of non-Abelian

isomorphic types of groups of order 2p, 3p, 5p, 7p, 11p, 13p where p < 4000. Also

considered are groups of order 2pq, 3pq, 5pq and 7pq. The primes p and q are distinct

primes with p < q.

1.4 DEFINITION OF THE CONCEPT OF ISOMORPHIC GROUPS

The concept of group isomorphism can be explained with chessboard that has four

plane symmetries. The identity, rotation r through about its centre, and the

reflections 1 2 q , q in its two diagonals form a group under composition whose

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multiplication is given in table 1 below.

Table 1.1: Four plane symmetries of a chessboard

e r 1 q 2 q

e e r 1 q 2 q

r r e 2 q 1 q

1 q 1 q 2 q

e r

2 q 2 q 1 q r e

It is easy to check that multiplication modulo eight makes the numbers 1,3,5,7 into a

group.

There is an apparent similarity between these two groups if we ignore their origins. In

each case the group has four elements, and these elements appear to combine in the

same manner. Only the way in which the elements are labeled distinguishes one table

from the other.

Label the first group G, the second G’, and the correspondence.

1, 3, 5, 7 1 2 e r q q ,

This correspondence is called an isomorphism between G and G’. It is a bijection and

it carries the multiplication of G to that of G’. Technically they are isomorphic in the

following sense.

Two groups G and G’ are isomorphic if there is a bijection from G to G’ which

satisfies xyxy for all x, yG. The function is called an isomorphism

between G and G’.

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Hence the isomorphism as a bijection implies that G and G’ must have the same order.

It sends the identity of G to that of G’. Isomorphism also preserves the order of each

element (Armstrong, 1988).

1.4.1 EXAMPLES

(i) The group of all real numbers with addition, (R,+), is isomorphic to the group of

all positive real numbers with multiplication ( + × ) R , .

Proof: Define f :(R,+) →( + ×) R , by f(x) = ex. For elements x, y in R then

f(x) = f(y) then ex = ey, so x = y. This implies that x ≠ y, then f(x) ≠ f(y) i.e., ex ≠ ey.

If r is an element o R+, then f(ln r) = r, where ln r belong to R showing that f is onto

R+ . Again, for elements x, y in R, we have

f(x + y) = ex+y = ex.ey = f(x)f(y).

Hence (R,+) is isomorphic to( + ×) R , .

(ii) Every cyclic group of infinite order is isomorphic to the additive group I of

integers

Proof: Consider the infinite cyclic group G generated by a and the mapping

n → an, n∈ I of I into G.

Now, this mapping is onto since any n in I is mapped to exactly one an.

Moreover, it is one-to one since if s > t we have s ↔ as and t ↔ at, then

as-t =1 and G would be finite. Hence if s ‡ t, then as ≠ at.

Finally, s + t ↔ as+t = as.at. Hence the mapping is an isomorphism, that is

I G.

(iii) The group Z of integers (with addition) is a subgroup of R, and the factor group

R/Z is isomorphic to the group S’ of complex numbers of absolute value 1

(with multiplication):

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R/Z S’

An isomorphism is given by

f(x+Z) = e2xi

for every x in R.

Proof: We only need to show that for any k ε Z, then f(x + k) = e2i(x +k) = e2xi +2iπk

= e2xi. e2ki = e2xi(Cos2πk + iSin2πk) = e2xi.

If x ≠ y then f(x + k) ≠ f(y + k), i.e., e2xi ≠ e2yi . Also for z ε R, then f(ln(z + k))

= eln(z + k) = z + k.

(iv) The Klein four-group is isomorphic to the direct product of two copies of

Z2 = Z/2Z and can therefore be written Z2xZ2. Another notation is D2, because

it is a dihedral group.

(v) Generalizing this, for all odd n, D2n is isomorphic with the direct product of

Dn and Z2.

1.4.2 PROPERTIES OF ISOMORPHIC GROUPS

(i) The Kernel of an isomorphism from (G,*) to (H, ʘ), is always {eG} where eG is the identity of the group (G,*).

(ii) If (G,*) is isomorphic to (H,ʘ), and if G is Abelian then so is H.

(iii) If (G,*) is a group that is isomorphic to (H,ʘ) [where f is the isomorphism], then if a belongs to G and has order n, then so does f(a)

(iv) If (G,*) is a locally finite group that is isomorphic to (H,ʘ), then (H,ʘ), is also locally finite.

We state mostly without proof certain fundamental results of group theory which we shall be needed:

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1.4.3 THEOREM (LAGRANGE’S THEOREM)

Let G be a group of finite order n, and H a subgroup of G. The order of H divides the

order of G.

1.4.4 THEOREM (CAUCHY’S THEOREM)

If p is a prime number and

p G

then G has an element of order p.

1.4.5 THEOREM (SYLOW’S FIRST THEOREM)

If pa is the highest power of a prime dividing the order of a group G, then G has at

least one subgroup of order p

1.4.6 DEFINITION 1

For any prime, p, we say that a group G is a p-group if every element x ,iny GG has order

pk, for some integer k

1.4.7 DEFINITION 2

Let G be a finite group of order n = pq, where (p,q) = 1. Then any subgroup of order

pm is called a Sylow p-subgroup of G.

1.4.8. DEFINITION 3

Let a be an element of a group G and e the identity element of G. The smallest

positive integer n such that an = e is called the order of a. The order of a group G,

written G is the cardinal number of elements of G. G is said to be finite or infinite

according as its order is finite or infinite (Kuku, 1980).

1.4.9 DEFINITION 4

Let G be a group and let a and b be elements of G then G contains both a and b .

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Other elements of G depends on the relation between a and b. The smallest subgroup

generated by a anb b is denoted by a,b . If ab = ba then G is said to be Abelian or

commutative. If ab≠ba then G is said to be non-Abelian or is said to be not

commutative.

1.4.10 THEOREM (SYLOW’S SECOND THEOREM)

All Sylow p-subgroups of a finite group G belonging to the same prime are conjugate

with one another in G.

1.4.11 THEOREM (SYLOW’S THIRD THEOREM)

Let r be the number of Sylow p-subgroups of G, then r is an integer of the form 1+kp

and r is a factor of the order of G.

1.4.12 THEOREM (A BASIS THEOREM FOR FINITE ABELIAN GROUPS)

Every finite Abelian group is a direct sum of primary cyclic groups.

1.4.13 THEOREM (ANOTHER BASIS THEOREM FOR FINITE ABELIAN GROUPS)

Every finite Abelian group A can be decomposed into a direct sum of cyclic groups.

1 2 1

….

m m ms A C C C

Where i m m 11 for all i= 1,2,…,s-1

1.4.14 THEOREM

If H and K are normal subgroups of G such that HK 1 then any element x ,oyf HG

commutes with any other element y of K.

PROOF:

For any xH and yK, consider the commentator

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1 1 1 1 1 1 z xyx y xyx y x yx y and notice from the first factorization

and the normality of K that y-1K, xy, yx,yGG -1 K zK.

Furthermore, since H is normal, we have from the second factorization that

xH yx y H zH 1 1 ,

Hence, we deduce that

zHK 1z 1

Whence

xy yx as asserted

1.4.15 PROPOSITION

Let G be a finite group and K any normal subgroup contained in the centre of the

group G. Then if G is non-Abelian the quotient group G/K cannot be cyclic.

PROOF:

Suppose

G/K K, tK,…, t K n1

Then for any x, yG we have

x=, ytu,Gy t v, s r

For some u,vK and thus

xy, y= tGsutrv = tk+ruv = tr+svu = trvtsu = y x , yG

(Since u, v permute with t).

This contradicts the non-Abelian hypothesis on G.

The problem of explicitly constructing all the groups of a given finite order has a long

and somewhat chequered history; its study was initiated by Cayley in 1864 when he

determined the groups of order at most 6. The aim is to determine a complete and

22

irredundant list of the groups of a given order: a representative of each isomorphism type present. It is usually comparatively easy to generate a complete list; the difficulty lies in the reduction to distinct isomorphism types (Hall M., 1976).

1.4.16 THEOREM (FROBENIUS)

Let H be a p-subgroup of order pa in G. Let K, of order pb, be the intersection of H and some other p-Sylow subgroup H’ of G such that no subgroup of G containing K and of order greater than pb is contained in any two p-Sylow subgroups. Then G must contain an element of order prime to p which permutes with K but does not permute with H.

1.4.17 REMARK

a) The subgroup K is a subgroup of maximum order common to both H and H’, it does not necessarily have maximal order among the intersections of any two p-Sylow subgroups.

b) There is a parallel theorem when p-Sylow subgroups H and H’ are both Abelian. In this case, every element of K is self-conjugate in the subgroup gp{H,H’}.

Thus if N is the greatest subgroup of G in which every element of K is self-conjugate, then N contains two and hence

1 + kp p-Sylow subgroups.

That is, N has order pam’(1+kp) where pam’ is the order of the greatest subgroup of the normalizer of H (of order pam) in which every element of K is self-conjugate. Thus, in this case, there is an element of order p which permutes with every element of K.

1.4.18 POLYNOMIAL:

A function of z of the form

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P(z) = a0 + a1z + a2z2 + … + anzn,

in which an ≠ 0 is called a polynomial of degree n in z.

1.4.19 THEOREM

Every polynomial of degree n (where n > 0) has at least one root and at most n roots (Mervin 1986).

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