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Algebra - Queen of Mathematics
Mathematical Notations
N - Set of all Natural numbers
Z - Set of all integers
W - Set of all whole numbers
Z⁺ - Set of all positive integers
Q - Set of all rational numbers
Q⁺ - Set of all positive rational numbers
Q* - Set of all non-zero rational numbers
R - Set of all real numbers
R⁺ - Set of all positive real numbers
R* - Set of all non-zero real numbers
C - Set of all Complex Numbers
C* - Set of all non-zero Complex numbers
1
Binary Operation
Let S be any non-empty set. An operation * is said to be a binary operation on S if:
∀ a, b ∈ S ⇒ a * b ∈ S
Examples:
+, ×, − are binary operations on β
+, ×, − are binary operations on β€
− is not a binary operation on β
+ is a binary operation on β*, β*, β*
− is a binary operation on β
Key Points:
A binary operation combines two elements of a set to produce another element of the same set
Closure property is essential for binary operations
2
Group Theory
Let G be a non-empty set and * be a binary operation on G. Then (G, *) is a group if:
Closure: ∀ a, b ∈ G ⇒ a * b ∈ G
Associativity: ∀ a, b, c ∈ G ⇒ (a * b) * c = a * (b * c)
Identity: ∃ e ∈ G such that a * e = e * a = a
Inverse: ∀ a ∈ G, ∃ a⁻¹ ∈ G such that a * a⁻¹ = a⁻¹ * a = e
Group Examples:
(β€, +) is a group
(β, +) is not a group
(β€, -) is not a group
(β, -) is not a group
(β, +) is a group
(β*, +), (β*, +), (β*, +) are not groups
(β*, ×), (β*, ×), (β*, ×) are groups
(β, -), (β, -), (β, -) are not groups
Set of even integers (β€β, +) is a group
Set of odd integers (β€β, +) is not a group
3
More Group Examples
Additional Group Examples:
The set of all cube roots of unity {1, Ο, Ο²} under multiplication is a group
The set of all fourth roots of unity {1, i, -1, -i} under multiplication is a group
The set of all nth roots of unity under multiplication is a group
The set of all n×n real number matrices under addition is a group
The set of all n×n real number matrices under multiplication is not a group (except non-singular matrices)
The set of 2×2 matrices of the form
$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$
with ad-bc ≠ 0 forms a group under multiplication
Problem 1:
If $G = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \mid a,b,c,d \in \mathbb{R} \right\}$ forms a group under addition, find identity and inverse elements.
A group is called simple if it has no proper normal subgroups.
Properties of Simple Groups:
Every group of prime order is simple
Simple ⇒ cyclic ⇒ Abelian
A group that has only improper subgroups is simple
15
Number Theoretic Theorems
Euler's Theorem
If n is a positive integer and a is an integer relatively prime to n, then:
aΟ(n) ≡ 1 (mod n)
where Ο(n) is Euler's totient function.
Problem:
Find the remainder when 750 is divided by 12.
Solution:
Ο(12) = 4, and gcd(7,12) = 1
74 ≡ 1 (mod 12) ⇒ (74)12 = 748 ≡ 1 (mod 12)
750 = 748·72 ≡ 1·49 ≡ 1 (mod 12)
So remainder is 1.
Fermat's Little Theorem
If p is a prime number and a is any integer not divisible by p, then:
ap-1 ≡ 1 (mod p)
Problem:
Find the remainder when 3100 is divided by 13.
Solution:
By Fermat's Little Theorem: 312 ≡ 1 (mod 13)
3100 = 396·34 ≡ 1·81 ≡ 3 (mod 13)
So remainder is 3.
16
Normal Subgroups
Normal Subgroup
Let G be a group and N be a subgroup of G. Then N is called a normal subgroup of G if:
∀ g ∈ G and ∀ n ∈ N ⇒ gng⁻¹ ∈ N
We denote this by N ◁ G.
Example:
G = {1, -1, i, -i}, N = {1, -1}
For g = i, g⁻¹ = -i
gNg⁻¹ = {i·1·(-i), i·(-1)·(-i)} = {1, -1} = N
So N is normal in G.
Equivalent Conditions for Normal Subgroup:
N is normal in G if and only if gNg⁻¹ = N for all g ∈ G
N is normal in G if and only if gN = Ng for all g ∈ G
N is normal in G if and only if every left coset is a right coset
Properties of Normal Subgroups:
Every subgroup of an Abelian group is normal
Every subgroup of a cyclic group is normal
The center of G, Z(G), is normal
If H is a subgroup of index 2, then H is normal
The intersection of two normal subgroups is normal
If N₁ and N₂ are normal subgroups, then N₁N₂ is also normal
17
Algebra - The Queen of Mathematics
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In elementary algebra, these symbols represent quantities without fixed values, known as variables.
Key Concepts Covered:
Groups: Algebraic structures with a single binary operation satisfying closure, associativity, identity, and invertibility
Binary Operations: Operations that combine two elements to form another element of the same set
Subgroups: Subsets of groups that are themselves groups under the same operation
Cyclic Groups: Groups that can be generated by a single element
Normal Subgroups: Subgroups that are invariant under conjugation by group elements
Cosets: Sets formed by multiplying a subgroup by a fixed element
Applications:
Group theory has applications in many areas including:
Cryptography and coding theory
Quantum mechanics and particle physics
Crystallography and material science
Music theory and arts
Computer science and algorithm design
This study material covers the fundamental concepts of abstract algebra that are essential for the PG TRB Mathematics examination.