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PG TRB Mathematics Examination

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PG TRB Maths - Algebra Study Material

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

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 ℝ
+, ×, − are binary operations on ℤ
− is not a binary operation on ℕ
+ is a binary operation on ℝ*, ℚ*, ℂ*
− is a binary operation on ℝ

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

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:

(ℤ, +) is a group
(ℕ, +) is not a group
(ℤ, -) is not a group
(ℕ, -) is not a group
(ℝ, +) is a group
(ℝ*, +), (ℂ*, +), (ℚ*, +) are not groups
(ℝ*, ×), (ℂ*, ×), (ℚ*, ×) are groups
(ℝ, -), (ℚ, -), (ℂ, -) are not groups
Set of even integers (ℤₑ, +) is a group
Set of odd integers (ℤₒ, +) is not a group

More Group Examples

Additional Group Examples:

The set of all cube roots of unity {1, ω, ω²} 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.

Solution:

Identity element: $I = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Inverse of $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $-A = \begin{bmatrix} -a & -b \\ -c & -d \end{bmatrix}$

Group Problems & Solutions

Problem 2:

If (ℤ, *) forms a group with operation defined by a*b = a + b + 2, find identity and inverse elements.

Solution:

We know: a*e = e*a = a

So: a + e + 2 = a ⇒ e = -2

For inverse: a*a⁻¹ = a⁻¹*a = e

a + a⁻¹ + 2 = -2 ⇒ a⁻¹ = -4 - a

Problem 3:

If (ℚ, ×) forms a group with operation defined by a×b = a + b - ab, find identity and inverse.

Solution:

a×e = a + e - ae = a ⇒ e - ae = 0 ⇒ e(1-a) = 0 ⇒ e = 0

For inverse: a×a⁻¹ = a + a⁻¹ - aa⁻¹ = 0

a⁻¹(1-a) = -a ⇒ a⁻¹ = a/(a-1) for a ≠ 1

Problem 4:

If G = {(x, x) | x ∈ ℝ*} forms a group under multiplication, find identity and inverse.

Solution:

Identity: (e, e) such that (x, x)(e, e) = (x, x) ⇒ xe = x ⇒ e = 1

So identity is (1, 1)

Inverse: (x, x)(y, y) = (1, 1) ⇒ xy = 1 ⇒ y = 1/x

So inverse is (1/x, 1/x)

Function Groups & Abelian Groups

Function Group Example:

If H = {f₁, f₂, f₃, f₄} forms a group under composition where:

f₁(x) = x
f₂(x) = -x
f₃(x) = 1/x
f₄(x) = -1/x

Then f₁ is the identity function, and each function has an inverse.

Commutative (Abelian) Group

A group G is Abelian if it satisfies the commutative property:

∀ a, b ∈ G ⇒ a * b = b * a

Abelian Group Examples:

(ℤ, +) is an infinite Abelian group
The set of all n×n matrices under addition is Abelian
The set of all n×n matrices under multiplication is not Abelian

Semi-groups & Monoids

Semi-group

A non-empty set G is a semi-group under binary operation if it satisfies:

Closure property
Associative property

Note: Every group is a semi-group, but the converse is not true.

Semi-group Examples:

(ℕ, +) is a semi-group but not a group
(ℤ, ×) is a semi-group
(ℤₙ, ×) modulo n is a semi-group

Monoid

A semi-group that satisfies the identity element property is called a monoid.

Monoid Examples:

(ℤ, ×) is a monoid
(ℤ, +) is a monoid
(ℕ, +) is not a monoid

Order of Groups & Elements

Order of a Group

The order of a group G, denoted o(G), is the number of distinct elements in G.

Examples:

If G = {1, 2, 3, 4}, then o(G) = 4
(ℤ, +) has infinite order

Residue Classes

The residue class modulo n is the set of all congruence classes from 0 to n-1:

ℤₙ = {[0], [1], [2], ..., [n-1]}

ℤₙ forms an Abelian group under addition modulo n.

Order of an Element

Let G be a group and a ∈ G. The smallest positive integer n such that aⁿ = e is called the order of a, denoted o(a).

Example:

If G = {1, ω, ω²} is a group under multiplication where ω³ = 1, then o(ω) = 3.

Properties of Groups

Key Properties of Groups:

In a group, the identity element is unique
The inverse of every element is unique
The inverse of an inverse element is the element itself: (a⁻¹)⁻¹ = a
Reverse law: (a * b)⁻¹ = b⁻¹ * a⁻¹
Left cancellation law: a*b = a*c ⇒ b = c
Right cancellation law: b*a = c*a ⇒ b = c
If G is a group and a,b ∈ G, then the equations ax = b and ya = b have unique solutions

Problem:

Find the solution of equation ax = b in S₃ where:

a = $\begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}$, b = $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{pmatrix}$

Solution: x = a⁻¹b = $\begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}$

Additional Properties:

If every element of a group has order 2, then the group is Abelian
If (a*b)² = a²*b² for all a,b in G, then G is Abelian
If G is Abelian, then (a*b)ⁿ = aⁿ*bⁿ for all integers n
If o(G) = 2n where n > 3, then G is non-Abelian
Every group of order ≤ 5 is Abelian
Every group of prime order is cyclic and hence Abelian

Special Elements & Subgroups

Idempotent Element

Let G be a group and a ∈ G. If a² = a, then a is called an idempotent element.

In every group, the only idempotent element is the identity element.

Periodic (Torsion) Group

A group is said to be a periodic group if every element of the group has finite order.

Examples:

G = {1, -1, i, -i} is periodic
G = {1, ω, ω²} is periodic

Subgroup

Let G be a group and H be a subset of G. If H forms a group under the binary operation of G, then H is called a subgroup of G.

Examples:

(ℤₑ, +) is a subgroup of (ℤ, +) where ℤₑ is the set of even integers
(ℤ, +) is a subgroup of (ℝ, +)
H = {1, -1} is a subgroup of G = {1, -1, i, -i} under multiplication

Subgroup Theorems & Properties

Subgroup Tests:

A non-empty subset H of a group G is a subgroup of G if ∀ a,b ∈ H ⇒ ab ∈ H and ∀ a ∈ H ⇒ a⁻¹ ∈ H
A non-empty subset H of a group G is a subgroup of G if ∀ a,b ∈ H ⇒ ab⁻¹ ∈ H
The identity element of a group and its subgroup are the same
If G is finite and H is a non-empty subset of G, then H is a subgroup if ∀ a,b ∈ H ⇒ ab ∈ H

Union and Intersection of Subgroups:

The union of two subgroups is not necessarily a subgroup
The union of two subgroups is a subgroup only if one is contained in the other
The intersection of two subgroups is always a subgroup
If H and K are two subgroups of G, then HK is a subgroup of G if and only if HK = KH

Order of Product Subgroups:

If H and K are finite subgroups of G, then:

o(HK) = o(H)o(K)/o(H∩K)

If o(H∩K) = 1, then o(HK) = o(H)o(K)

Center & Cyclic Groups

Center of a Group

Let G be a group. The center of G, denoted Z(G), is defined as:

Z(G) = {x ∈ G | xa = ax ∀ a ∈ G}

The center of a group is always a subgroup of G.

Cyclic Group

A group G is called cyclic if there exists an element a ∈ G such that every element of G can be expressed as a power of a:

G = {aⁿ | n ∈ ℤ}

We write G = ⟨a⟩ and say that a is a generator of G.

Examples:

(ℤ, +) is cyclic with generators 1 and -1
G = {1, -1, i, -i} is cyclic with generators i and -i
If o(G) = p² where p is prime, then G is cyclic

Monogenic Cyclic Group

A cyclic group that has only one generator is called a monogenic cyclic group.

Example:

G = {1, -1} is monogenic with -1 as the only generator.

Properties of Cyclic Groups

Key Properties of Cyclic Groups:

The order of a cyclic group is the same as the order of its generator
A cyclic group can have several generators
If a is a generator of a cyclic group G, then a⁻¹ is also a generator of G
Every cyclic group is Abelian, but the converse is not true
Every subgroup of a cyclic group is cyclic
Every subgroup of a cyclic group is Abelian
A cyclic group of order n has exactly φ(n) generators, where φ is Euler's totient function
An infinite cyclic group has exactly two generators
Every group of prime order is cyclic

Problem:

If G = {1, ω, ω²} is a cyclic group, find its generators.

Solution:

ω¹ = ω, ω² = ω², ω³ = 1 ⇒ o(ω) = 3

(ω²)¹ = ω², (ω²)² = ω⁴ = ω, (ω²)³ = ω⁶ = 1 ⇒ o(ω²) = 3

So generators are ω and ω².

Problem:

Find the number of generators of a cyclic group of order 15.

Solution: φ(15) = 15(1-1/3)(1-1/5) = 15×(2/3)×(4/5) = 8

So there are 8 generators.

Cosets & Lagrange's Theorem

Coset of a Subgroup

Let G be a group and H be a subgroup of G. Then for any a ∈ G:

The set Ha = {ha | h ∈ H} is called a right coset of H
The set aH = {ah | h ∈ H} is called a left coset of H

Example:

G = ℤ = {0, ±1, ±2, ...} under addition

H = 2ℤ = {0, ±2, ±4, ...}

For a = 1 ∈ G, H+1 = {±1, ±3, ±5, ...} is a coset of H

Properties of Cosets:

The union of all cosets is the whole group G
If a ∈ H, then Ha = H
Any two right (or left) cosets have the same number of elements
Any two distinct cosets are disjoint
The number of right cosets equals the number of left cosets

Lagrange's Theorem

Let G be a finite group and H be a subgroup of G. Then the order of H divides the order of G:

o(H) | o(G)

Index & Simple Groups

Index of a Subgroup

Let G be a group and H be a subgroup of G. The number of distinct cosets of H in G is called the index of H in G, denoted [G:H] or i_G(H).

[G:H] = o(G)/o(H)

Problem 1:

If G is a group of order 30 and H is a subgroup of order 10, find the index of H.

Solution: [G:H] = 30/10 = 3

Problem 2:

If G is a group of order 10 and the index of a subgroup is 5, find the order of the subgroup.

Solution: [G:H] = o(G)/o(H) ⇒ 5 = 10/o(H) ⇒ o(H) = 2

Index Product Theorem:

If H and K are subgroups of G with K ⊆ H, then:

[G:K] = [G:H]·[H:K]

Simple Group

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

Number Theoretic Theorems

Euler's Theorem

If n is a positive integer and a is an integer relatively prime to n, then:

a^φ(n) ≡ 1 (mod n)

where φ(n) is Euler's totient function.

Problem:

Find the remainder when 7⁵⁰ is divided by 12.

Solution:

φ(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:

a^p-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.

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

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.

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ALGEBRA FLIP BOOK

PG TRB Mathematics

Unit 1: Algebra Study Material

Algebra - Queen of Mathematics

Mathematical Notations

Binary Operation

Examples:

Key Points:

Group Theory

Group Examples:

More Group Examples

Additional Group Examples:

Problem 1:

Group Problems & Solutions

Problem 2:

Problem 3:

Problem 4:

Function Groups & Abelian Groups

Function Group Example:

Commutative (Abelian) Group

Abelian Group Examples:

Semi-groups & Monoids

Semi-group

Semi-group Examples:

Monoid

Monoid Examples:

Order of Groups & Elements

Order of a Group

Examples:

Residue Classes

Order of an Element

Example:

Properties of Groups

Key Properties of Groups:

Problem:

Additional Properties:

Special Elements & Subgroups

Idempotent Element

Periodic (Torsion) Group

Examples:

Subgroup

Examples:

Subgroup Theorems & Properties

Subgroup Tests:

Union and Intersection of Subgroups:

Order of Product Subgroups:

Center & Cyclic Groups

Center of a Group

Cyclic Group

Examples:

Monogenic Cyclic Group

Example:

Properties of Cyclic Groups

Key Properties of Cyclic Groups:

Problem:

Problem:

Cosets & Lagrange's Theorem

Coset of a Subgroup

Example:

Properties of Cosets:

Lagrange's Theorem

Index & Simple Groups

Index of a Subgroup

Problem 1:

Problem 2:

Index Product Theorem:

Simple Group

Properties of Simple Groups:

Number Theoretic Theorems

Euler's Theorem

Problem:

Fermat's Little Theorem

Problem: