ALGEBRA FLIP BOOK

PG TRB Mathematics

Unit 1: Algebra Study Material

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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 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

2

Group Theory

Let G be a non-empty set and * be a binary operation on G. Then (G, *) is a group if:

1. Closure: ∀ a, b ∈ G ⇒ a * b ∈ G
2. Associativity: ∀ a, b, c ∈ G ⇒ (a * b) * c = a * (b * c)
3. Identity: ∃ e ∈ G such that a * e = e * a = a
4. 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

3

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

4

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)

5

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

6

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

7

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.

8

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

9

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

10

Subgroup Theorems & Properties

Subgroup Tests:

1. 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
2. A non-empty subset H of a group G is a subgroup of G if ∀ a,b ∈ H ⇒ ab⁻¹ ∈ H
3. The identity element of a group and its subgroup are the same
4. 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)

11

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.

12

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.

13

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)

14

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

15

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.

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.