1. The Beauty of
Abstract Algebra
Overview
- What is Algebra ?
-What is Abstract Algebra ?
-Group
-Some definitions( Subgroup, cyclic group , order of
group ,cosets)
-Beauty of Abstract Algebra
-Dihedral groups and its real life applications
-Dihedral groups in nature
-Other applications of Abstract Algebra
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2. Algebra:
In mathematics , we use symbols to represent mathematical
objects(numbers , matrices, transformations,…) and the study
of the rules for combining these symbols under some
operation is called algebra.
Eg. (𝑎 + 𝑏)2= 𝑎2 + 𝑏2 + 2𝑎𝑏
Linear Algebra
Advanced Algebra
Abstract Algebra
Elementary algebra
Commutative Algebra
Algebra
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3. Abstract algebra:
It deals with the study of algebraic systems or structures
with one or more mathematical operations associated
elements with an identifiable pattern, differing from the
usual number systems.
In this mainly we study –
Rings
Vectorspaces
Groups
Fields
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4. GROUP-
Definition : A group is a set G , together with a binary operation ∗,
such that the following hold:
1. (Associativity): (a ∗ b) ∗ c = a ∗ (b ∗ c) ∀a, b, c ∈ G.
2. (Existence of identity): ∃ e ∈ G such that a ∗ e = e ∗ a = a ∀a ∈ G.
3. (Existence of inverses): Given a ∈ G, ∃ b ∈ G such that a ∗ b = b ∗ a = e
Example: The set of integers z under ordinary addition is a
group.
Order of a Group: The number of elements of a group (finite or
infinite) is called its order. We will use |G| to denote the order of G.
Example: The set 𝑧8 = {0,1,2,3,4,5,6,7} is a group under ⊕8with
order 8. PP14
5. Subgroup
Definition: If a subset H of a group G is itself a group under the
operation of G, we say that H is a subgroup of G.
Example: H={0,2,4,6,8} is a subgroup of 𝑍10.
CYCLIC GROUP
A group G is called cyclic if there is an element a in G such that
G={𝑎𝑛| 𝑛ϵ𝑧}. Such an element a is called a generator of G.
Example: Z under ordinary addition is cyclic. Both 1 and -1 are
generators.
Cosets of H in G :
Let G be a group and let H be a subgroup of G. For any 𝑎ϵG, the set
{𝑎ℎ|ℎϵH}is denoted by aH is called the left coset of H in G containing a.
Analogously, Ha is called the right coset of H in G containing a. PP14
6. Beauty of Abstract Algebra
Theorem: If G is a finite group and H is a subgroup of G, then |H|
divides |G|.
The beauty embedded in this theorem is that , we didn’t even know what
kind of elements do group G have whether they are numbers or matrices
or any other mathematical object, what is the binary operation of G but
still we can prove that G can have subgroups whose order is a divisor of
|G|.
Result: Groups of prime order are cyclic.
Similarly , we didn’t even know anything about the elements and binary
operation of G but still we can claim that if |G| is prime the G is cyclic.
With out knowing much about any algebraic systems (groups , rings,…)PP14
7. Dihedral Group :
For every integer 𝑛 ≥ 3, the regular polygon with n sides has a group
of symmetries, symbolized by 𝐷𝑛. These groups are called the dihedral
groups .
Real life applications of Dihedral groups –
Corporation logos are rich sources of dihedral symmetry.
Chrysler’s logo has 𝐷5 as a symmetry group, and that of
Mercedes- Benz has 𝐷3 .
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8. Dihedral groups in nature –
1.) The phylum Echinodermata contains many sea animals (such as
starfish
,sea cucumbers, feather stars, and sand dollars) that exhibit patterns
with 𝐷5 symmetry.
2.) Snowflakes has 𝐷6 symmetry.
Starfish sea cucumbers sand dollars
Snowflake PP14
9. Some other applications of Abstract Algebra –
1.) Chemists classify molecules according to their symmetry .For eg.
The
symmetry group of a pyramidal molecule such as ammonia (𝑁𝐻3) is
D3 .
2.) Ring Theory has been well-used in cryptography and many others
computer vision tasks.
3.) Group theory algorithms are used to solve Rubik’s cube.
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