FINITE GROUP AND FINITE GEOMETRICS
BY
ABDULRASHEED, SHEIKH ABDULWALIY
(SCI21MTH028)
A PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS,
FACULTY OF SCIENCE, FEDERAL UNIVERSITY LOKOJA, KOGI
STATE
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
AWARD OF BACHELOR OF SCIENCE (B.SC.) HONS IN MATHEMATICS
FEDERAL UNIVERSITY LOKOJA.
MARCH, 2025
DECLARATION
I hereby declare that this project work is the product of my research effort,
undertaken by my supervision of Dr. Kehinde and has not been presented and
will not be presented elsewhere elsewhere for the award of any degree or
certificate. All sources has been duly acknowledged.
………………………………
………………….
Name of Student
Date
………………….
Signature
CERTIFICATION
This is to certify that the project titled “Finite Group and Finite Geometrics”
was carried out by Abdulrasheed Sheikh Abdulwaliy with matric number
SCI21MTH028 under the supervision of Dr. Kehinde. I have neither copy nor
duplicated the work of any other(s). all works used have been duly cited and
acknowledged.
….……………………...
Dr. Kehinde
Project Supervisor
Signature/Date
Prof. M. A. Ibrahim
………………………….
Head of Department
Signature/Date
…………………………..
………………………….
External Examiner
Signature/Date
n
ACKNOWLEDGEMENT
First and foremost, all thanks and appreciation goes to God almighty for the gift
of life and the strength to carry out this research work successfully.
I am highly indebted to my supervisor Dr. Kehinde for his excellent directions,
constructive suggestions, detailed corrections, support and encouragement
which played an enormous role to the success of this project.
I am most grateful to all departmental lecturers who has deeply impacted my
life. I pray that God almighty will guild your proceeds.
I owe a debt of gratitude to my parents Mr. and Mrs. Abdulwaliy for their
prayers, support and care, and my siblings. God bless you all.
Finally, to my friends, I can never forget you guys, may the Lord Almighty
bless you all and I say thank you all for all your words of encouragement.
DEDICATION
This project work is dedicated to the Almighty God and to my family.
ABSTRACT
TABLE OF CONTENT
i DECLARATION ……………………………………………………………
ii CERTIFICATION …………………………………………………………..
iii ACKNOWLEDGEMENT ………………………………………………….
iv DEDICATION ……………………………………………………………..
v ABSTRACT ………………………………………………………………..
1. INTRODUCTION
1.1 Brief history on finite group and finite geometrics …………………….
1.2 Statement of the problem ………………………………………………
1.3 Aims and Objectives ……………………………………………………
1.4 Scope and limitation of the study……………………………………….
1.5 Definition of Terms …………………………………………………….
1.5.1 Group ………………………………………………………………...
1.5.2 Subgroup ……………………………………………………………..
1.5.3 Finite group ………………………………………………………….
1.5.4 Permutation group……………………………………………………
1.5.5 Cyclic group………………………………………………………….
1.5.6 Geometrics …………………………………………………………..
1.5.7 Finite Geometrics ……………………………………………………
1.5.8 Finite projective plane………………………………………………..
1.5.9 Finite affine plane…………………………………………………….
1.5.10 Group homomorphism………………………………………………..
1.5.11 Group isomorphism ………………………………………………….
1.5.12 Group automorphism ………………………………………………...
2. LITERATURE REVIEW
3. METHODOLOGY
3.1 Finite Group ………………………………………………………….
3.2 Finite Geometrics …………………………………………………….
3.3 Relationship between Finite Group and Finite Geometrics ………….
4. RESULT AND DISCUSSION
4.1 Finite order ……………………………………………………
4.2 Geometrics ………..……………………………………………………
4.3 The isomorphism of a group …………………………………………….
5. SUMMARY, CONCLUSION AND RECOMMENDATION
5.1. Summary ……………………………………………………………….
5.2. Conclusion ……………………………………………………………..
5.3. Recommendations ……………………………………………………..
REFERENCES
CHAPTER ONE
1 INTRODUCTION
1.1 BRIEF HISTORY OF FINITE GROUP AND FINITE
GEOMETRICS
Early beginnings (19th century):
 Group Theory: The study of groups began with Evariste Galois in
the 1830s, who introduced the concept of groups in the context of
solving polynomial equations.
 Geometry: Finite geometrics emerged from the study of projective
geometry, which was developed by mathematicians such as JeanVictor Poncelet and Julius Plucker in the early 19th century.
Development of Finite Group Theory (Late 19th-Early 20th
centuries):
 Sylow’s Theorems: Peter Sylow introduced his famous theorems
in 1872, which provided a fundamental understanding of finite
groups.
 Frobenius and Burnside: Ferdinand Frobenius and William
Burnside made significant contributions to the development of
finite group theory in the late 19th and early 20th centuries.
Emergence of finite geometrics (Early 20th century):
 Veblen and Young: Oswald Veblen and John Wesley Young
introduced the concept of finite projective geometrics in the early
20th century.
 Combinatorial Approach: The combinatorial approach to finite
geometrics emerged in the 1930s, with the work of mathematicians
such as Marshall Hall and Gordon Pall.
Connections between Finite Groups and Finite Geometrics (Mid20th Century):
 Tits and Suzuki: Jacques Tits and Michio Suzuki discovered
connections between finite groups and finite geometrics in the
1950s and 1960s.
 Classifications of Finite Simple Groups: The classification of
finite simple groups, completed in the 1980s, relied heavily on the
connections between finite groups and finite geometrics.
Modern Developments (Late 20th century-Present):
 Computational methods: The development of computational
methods and software packages has enabled researchers to study
finite groups and finite geometrics in greater depth.
 Applications and Interaction: Finite groups and finite geometrics
have found applications in various areas, including coding theory,
cryptography and computer science.
1.2 STATEMENT OF THE PROBLEM
1. Group actions: Examine how finite groups act on finite geometrics,
such as projective spaces and affine spaces.
2. Geometric interpretations: Explore geometric interpretations of
group theoretic concepts, such as conjugacy classes and character
theory.
3. Connections to combinatorics: Investigate connections between
finite groups, finite geometries, and combinatorial structures, such as
graphs and designs.
4. Applications: Examine applications of finite groups and finite
geometries in areas like coding theory, cryptography, and computer
science.
1.3 AIMS AND OBJECTIVES
AIMS:
1. Investigate Connections: Explore the relationships between finite
groups and finite geometries.
2. Advance Understanding: Advance our understanding of the interplay
between group theory and geometry.
3. Identify Applications: Identify potential applications of finite groups
and finite geometries in coding theory, cryptography, and computer
science.
OBJECTIVES:
PRIMARY OBJECTIVES:
1. Analyze Group Actions: Examine how finite groups act on finite
geometries, such as projective spaces and affine spaces.
2. Develop Geometric Interpretations: Develop geometric
interpretations of group theoretic concepts, such as conjugacy classes
and character theory.
3. Investigate Combinatorial Connections: Investigate connections
between finite groups, finite geometries, and combinatorial structures,
such as graphs and designs.
SECONDARY OBJECTIVES:
1. Explore Coding Theory Applications: Examine how finite groups
and finite geometries can be applied in coding theory.
2. Investigate Cryptographic Applications: Investigate how finite
groups and finite geometries can be used in cryptography.
3. Identify Computer Science Applications: Identify potential
applications of finite groups and finite geometries in computer
science.
1.4 SCOPE AND LIMITATIONS OF STUDY
SCOPE OF STUDY:
1. Group Theory: Finite groups are a fundamental area of study in
group theory, which is a branch of abstract algebra.
2. Geometry and Symmetry: Finite geometric groups are essential in
understanding the symmetries of geometric objects, which is crucial in
various areas of mathematics and computer science.
3. Computer Science: Finite groups and finite geometric groups have
numerous applications in computer science, including cryptography,
coding theory, computer graphics, and computer vision.
4. Physics and Engineering: Finite groups and finite geometric groups
are used to describe the symmetries of physical systems, which is
essential in physics and engineering.
LIMITATIONS OF STUDY:
1. Abstract Nature: Finite groups and finite geometric groups are
abstract mathematical structures, which can make them challenging to
study and visualize.
2. Complexity: As the size of the group increases, the complexity of the
group structure and its representations can become overwhelming.
3. Computational Challenges: Computing with finite groups and finite
geometric groups can be computationally intensive, especially for large
group.
4. Interdisciplinary Challenges: Applying finite groups and finite
geometric groups to real- world problems often requires
interdisciplinary knowledge and collaboration.
1.5 DEFINITION OF TERM
1.5.1 GROUP
If G is a non-empty set and “⋆” is the binary operation defined on G such
that the following laws or axioms are satisfied then, (G, ⋆) is called a
group.
Let “G” be a non-empty set and “⋆” be a binary operation on G such that
1. Closure law
For a, b ∈ G, a ⋆ b ∈ G
2. Associative law
a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c for all a, b, c ∈ G
3. Identity element
There is an element e ∈ G such that a ⋆ e = e ⋆ a = a for all a ∈ G; where e
is the identity element
4. Inverse law
For each a ∈ G, there exists an element b ∈ G such that a ⋆ b = b ⋆ a = e,
where b = a-1 is the inverse element of a.
1.5.2 SUBGROUP
Let (G, *) be a group structure and let S be a subset of G then S is said to
be a subgroup of G if (S, *) is a group structure and if and only if it
follows the properties given below:
1. Binary Structure: ab ∈ S for every a, b ∈ S.
2. Existence of Identity: Suppose e’ ∈ S such that e’a = a = ae’ for all a
∈ S.
3. Existence of Inverse: For all a ∈ S, there exists a−1 ∈ S such that aa−1 =
e = a−1a.
1.5.3 FINITE GROUP
In abstract algebra, a finite group is a group whose underlying set is
finite. Important examples of finite groups include cyclic group and
permutation group.
1.5.4 PERMUTATION GROUP
The symmetric group Sn on finite set of n symbols is the group whose
elements are all the permutations of the n symbols, and whose group
operation is the composition of such permutations, which are treated
as bijective functions from the set of symbols to itself. Since there are n!
possible permutations of a set of n symbols, it follows that the order (the
number of elements) of the symmetric group Sn is n!.
1.5.5 CYCLIC GROUP
A cyclic group ℤn is a group all of whose elements are powers of a
particular element a where an = a0 = e, the identity. A typical realization
of this group is as the complex nth roots of unity. Sending a to a primitive
root of unity gives an isomorphism between the two. This can be done
with any finite cyclic group.
1.5.6 GEOMETRICS
Geometry is a branch of mathematics concerned with properties of space
such as the distance, shape, size, and relative position of figures.
1.5.7 FINITE GEOMETRICS
A finite geometry is any geometric system that has only a finite number
of points.
1.5.6 FINITE PROJECTIVE PLANE
A projective plane geometry is a nonempty set X (whose elements are
called "points"), along with a nonempty collection L of subsets of X
(whose elements are called "lines"), such that:
1. For every two distinct points, there is exactly one line that contains
both points.
2. The intersection of any two distinct lines contains exactly one point.
3. There exists a set of four points, no three of which belong to the same
line.
1.5.7 FINITE AFFINE PLANE
An affine plane geometry is a nonempty set X (whose elements are called
"points"), along with a nonempty collection L of subsets of X (whose
elements are called "lines"), such that:
1. For every two distinct points, there is exactly one line that contains both
points.
2. Playfair’s axiom: Given a line
exists exactly one line
and a point
containing
not on
, there
such that
3. There exists a set of four points, no three of which belong to the same
line.
1.5.8 GROUP HOMOMORPHISM
Let G and H be groups. A group homomorphism from G to H is a
function φ : G → H such that
φ(ab) = φ(a)φ(b)
for all a, b ∈ G, or to be more precise, such that φ(a ∗ b) = φ(a) × φ(b) for
all a, b ∈ G, where ∗ is the operation on G and × is the operation on H.
The kernel of φ is the set
Ker(φ) = φ −1 (e) = {a ∈ G ∣ φ(a) = e}
where e = eH is the identity in H, and the image (or range) of φ is
Image(φ) = φ(G) = {φ(a) ∣ a ∈ G}
1.5.9 GROUP ISOMORPHISM
For two groups (G,+) and (G’,*) a mapping f: G → G’ is called an
isomorphism if

f is one-one

f is onto

f is homomorphism i.e. f(a + b) = f(a) * f(b) ∀ a, b ∈ G.
In brief, a bijective homomorphism is an isomorphism.
1.5.13 GROUP AUTOMORPHISM
For a group (G,+), a mapping f : G → G is called automorphism if
f is one-one.
f homomorphic i.e. f(a +b) = f(a) + f(b) ∀ a, b ∈ G.
CHAPTER TWO
2. LITERATURE REVIEW
Finite groups and finite geometries are closely related areas of mathematics
that have garnered significant attention in recent years.
Finite Groups
Finite groups are mathematical structures consisting of a finite set of
elements with a binary operation that satisfies certain properties. The study
of finite groups involves understanding their properties, classification and
applications.
- Classification: Finite simple groups can be classified into four types:
- Commutative groups of prime order: These groups have prime order
and are isomorphic to the cyclic group ℤp.
- Alternating groups: These groups are simple and can be represented as
An with n ≥ 5.
- Groups of Lie type: These groups are related to Lie algebras and are
used in various areas of mathematics and physics.
- Sporadic groups: These are 26 exceptional groups that don't fit into the
other categories.
Finite Geometries
Finite geometries, on the other hand, deal with geometric structures that have
a finite number of points, lines and planes. These structures have
applications in various fields, including:
- Experimental design: Finite geometries are used in the design of
experiments, particularly in the construction of combinatorial designs.
- Information security: Finite geometries are used in cryptography and
coding theory to develop secure encryption algorithms.
- Particle physics: Finite geometries are used to describe the symmetries of
particles in physics.
Some notable researchers in the field of finite geometry include:
- John Bamberg: Known for his work on generalized quadrangles and their
symmetries.
- Gordon Royle: Contributed to the classification of ovoids in finite
projective 3-space.
- Alice Devillers: Worked on the theory of buildings and their applications
to group theory.
CHAPTER THREE
3. METHODOLOGY