vii
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xii
1 INTRODUCTION
1
1.1
Introduction
1
1.2
Research Background
2
1.3
Problem Statements
3
1.4
Research Objectives
4
1.5
Scope of the Study
4
1.6
Significance of Findings
4
1.7
Research Methodology
5
1.8
Thesis Organization
6
2 LITERATURE REVIEW
8
2.1
Introduction
8
2.2
Definitions and Notations
8
2.3
Conjugacy Classes of a Finite Group
10
2.4
Conjugacy Class Sizes of a Finite Group
12
viii
2.5
Graphs Related to Conjugacy Classes
14
2.6
The Classification of 2-Generator 2-Groups of Class Two
16
2.7
Groups, Algorithms and Programming (GAP) Software
18
2.8
Conclusion
18
3 GAP CODING
19
3.1
Introduction
19
3.2
Groups of Type 1
19
3.3
Groups of Type 2
23
3.4
Groups of Type 3
25
3.5
Results Using GAP
27
3.6
Conclusion
27
4 CONJUGACY CLASSES OF 2-GENERATOR 2-GROUPS
OF CLASS TWO
31
4.1
Introduction
31
4.2
Preliminary Results
31
4.3
The Computation of the Conjugacy Classes
39
4.4
Conclusion
46
5 CONJUGACY CLASS SIZES OF 2-GENERATOR 2GROUPS OF CLASS TWO
47
5.1
Introduction
47
5.2
Preliminary Results
47
5.3
Conjugacy Class Sizes
52
5.4
Conclusion
56
6 GRAPH RELATED TO CONJUGACY CLASS SIZES OF
2-GENERATOR 2-GROUPS OF CLASS TWO
57
6.1
Introduction
57
6.2
Definitions and Notations
57
6.3
Preliminary Results
60
6.4
On Graph ΓG For Some Finite 2-Groups
62
6.5
Conclusion
66
ix
7 CONCLUSION
68
7.1
Summary of the Research
68
7.2
Suggestion for Future Research
69
7.2.1
Commuting Conjugacy Class Graph of a Group
70
7.2.2
Preliminary Results
70
REFERENCES
Appendices A - B
75
79 - 109
x
LIST OF TABLES
TABLE NO.
2.1
TITLE
The lower bounds and the exact number of conjugacy
classes for a p-group of nilpotency class two with |G| = pp
3.1
PAGE
13
The order of the group G, |G| , the order of the center
of group G, |Z (G)| , the order of the derived subgroup
G, |G0 | and the number of the conjugacy classes of G,
clG for Type 1 up to order 4096
3.2
28
The order of the group G, |G| , the order of the center
of group G, |Z (G)| , the order of the derived subgroup
G, |G0 | and the number of the conjugacy classes of G,
clG for Type 2 up to order 4096
3.3
29
The order of the group G, |G| , the order of the center
of group G, |Z (G)| , the order of the derived subgroup
G, |G0 | and the number of the conjugacy classes of G,
clG for Type 3 up to order 4096
30
5.1
Cayley Table of D4
49
5.2
Character Table of D4
51
7.1
The completeness, connectivity, diameter, chromatic
number and probability of commuting conjugacy classes
of 2-generator 2-groups of class two
73
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
The directed graph
14
2.2
The undirected graph
14
6.1
The graph related to conjugacy classes of D4
61
xii
LIST OF SYMBOLS
1
−
Identity element
hai
−
Cyclic subgroup generated by a
Cn
−
Cyclic group of order n
C(a)
−
Centralizer of a in G
cl(a)
−
Conjugacy class of a
clG
−
Number of conjugacy classes
Dn
−
Dihedral group of order 2n
d(ΓG )
−
Diameter of a graph G
d(x, y)
−
Distance between x and y in a graph G
E(ΓG )
−
Edge set of a graph G
|G|
−
Order of the group G
G0
−
Commutator subgroup of G
|G : H|
−
Index of the subgroup H in the group G
G/H
−
Factor group
G×H
−
Direct product of G and H
GoH
−
Semidirect product of G and H
G∼
=H
−
G is isomorphic to H
H≤G
−
H is a subgroup of G
H /G
−
H is a normal subgroup of G
Ker α
−
Kernel of the homomorphism α
N
−
Set of natural numbers
n(ΓG )
−
Number of connected components of G
P (G) −
Commutativity degree
Qn
−
Quaternion group of order 2n
V (ΓG )
−
Vertex set of a graph G
ω (ΓG )
−
Clique number of a graph G
|x|
−
Order of the element x
xiii
[x, y]
− The commutator of x and y
hX| Ri
−
χ (ΓG )
− Chromatic number of a graph G
Z
Groups presented by generators X and relators R
− Set of integers
Z/nZ
− Integers modulo n
Z (G)
− Center of the group G
ΓG
− Graph related to conjugacy classes of a group G
γG
− Graph related to commuting conjugacy classes of a group
G