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Groups

Below is a complete, clear, and exam-oriented explanation of Groups in Detail — ideal for BCA/MCA/Engineering/Discrete Mathematics and Abstract Algebra exams.

⭐ GROUPS – DETAILED DISCUSSION

A group is one of the most fundamental structures in abstract algebra.
It captures the idea of symmetry, operations, and invertibility.

Used heavily in:

✔ Mathematics
✔ Computer Science
✔ Cryptography
✔ Robotics
✔ Physics
✔ Automata Theory

⭐ 1. Definition of a Group

A set G with a binary operation (*) is called a group if it satisfies four properties:

1️⃣ Closure

For all ( a, b \in G ):

[
a * b \in G
]

2️⃣ Associativity

For all ( a, b, c \in G ):

[
(a * b) * c = a * (b * c)
]

3️⃣ Identity Element

There exists an element ( e \in G ) such that for all ( a \in G ):

[
e * a = a * e = a
]

4️⃣ Inverse Element

For every element ( a \in G ), there exists a unique ( a^{-1} ) (inverse of a) such that:

[
a * a^{-1} = a^{-1} * a = e
]

⭐ 2. Types of Groups

A. Finite and Infinite Groups

✔ Finite Group

Has finite number of elements (called order).
Example:
[
(\mathbb{Z}_7, +) \quad \text{order } 7
]

✔ Infinite Group

Example:
[
(\mathbb{Z}, +) \quad \text{infinite}
]

B. Abelian (Commutative) Group

If:

[
a * b = b * a \quad \forall a,b \in G
]

Then the group is Abelian.

Examples:

  • ((\mathbb{Z}, +))
  • ((\mathbb{R}-{0}, \times))

C. Non-Abelian (Non-Commutative) Group

If commutative property fails, the group is non-abelian.

Examples:

  • Symmetric group (S_n) for (n \ge 3)
  • Matrix groups (GL(n,\mathbb{R}))

⭐ 3. Examples of Groups

✔ Example 1: Integers under addition

Set: (\mathbb{Z})
Operation: +
Identity = 0
Inverse = -a

→ Abelian group

✔ Example 2: Non-zero real numbers under multiplication

Set: (\mathbb{R} – {0})
Operation: ×
Identity = 1
Inverse = 1/a

→ Abelian group

✔ Example 3: Symmetric Group (S_3)**

Set of all permutations of 3 symbols (6 elements).
Non-abelian.

Used in cryptography and computer algorithms.

✔ Example 4: 2×2 Invertible Matrices

[
GL(2,\mathbb{R})
]

Operation: matrix multiplication
Non-abelian group.

⭐ 4. Examples That Are NOT Groups

✘ Natural numbers under addition

No inverses → not a group.

✘ Integers under multiplication

Most numbers have no multiplicative inverse → not a group.

✘ Zero element in multiplication

0 has no inverse → not a group.

⭐ 5. Order of a Group and Element

Group order:

Number of elements in G
denoted |G|

Order of an element a:

Smallest positive integer n such that:

[
a^n = e
]

If no such n exists → order is infinite.

⭐ 6. Subgroups

A subset H of G is a subgroup if:

  1. Non-empty
  2. Closed under operation
  3. Closed under inverses

Short test (Subgroup Test):

[
a,b \in H \Rightarrow ab^{-1} \in H
]

Example:
Even integers under addition form a subgroup of ((\mathbb{Z},+)).

⭐ 7. Cyclic Groups

A group generated by a single element.

[
G = \langle a \rangle = {a^n : n \in \mathbb{Z}}
]

Example:

  • ((\mathbb{Z}, +)) cyclic generated by 1.
  • ((\mathbb{Z}_n, +)) cyclic generated by 1.

Cyclic groups are always abelian.

⭐ 8. Cosets

Given subgroup H of G and element a in G:

Left coset:
[
aH = {ah : h \in H}
]

Right coset:
[
Ha = {ha : h \in H}
]

Used in Lagrange’s Theorem and group partitioning.

⭐ 9. Lagrange’s Theorem (Very important)

In a finite group G:

Order of subgroup H divides order of group G.

[
|H| \mid |G|
]

Consequences:

  • Order of any element divides |G|.
  • Groups of prime order are cyclic.

⭐ 10. Homomorphisms

A function ( f : G \to H ) is a group homomorphism if:

[
f(a * b) = f(a) * f(b)
]

Special maps:

  • Isomorphism (structure-preserving bijection)
  • Automorphism (isomorphism from G to itself)
  • Endomorphism (map from G to itself)

⭐ 11. Group Table (Cayley Table)

Used to check closure and structure for small finite groups.

Rows and columns list group elements.
Entry (i, j) = product of row i and column j.

⭐ 12. Applications of Group Theory

✔ Computer Science

  • Automata (strings form monoids/groups)
  • Cryptography (RSA, Diffie–Hellman, ECC)
  • Symmetry in computational problems
  • Hashing, automorphisms

✔ Mathematics

  • Number theory
  • Geometry
  • Symmetry
  • Galois theory

✔ Physics

  • Quantum mechanics
  • Rotational symmetry
  • Conservation laws

✔ Chemistry

  • Molecular symmetry

✔ Robotics

  • Rotation groups (SO(3))

⭐ Exam-Oriented Summary

Group Definition:

Set with operation satisfying closure, associativity, identity, and inverses.

Types:

  • Finite, Infinite
  • Abelian, Non-Abelian
  • Cyclic
  • Symmetric groups
  • Matrix groups

Concepts:

  • Identity
  • Inverse
  • Order
  • Subgroups
  • Cosets
  • Lagrange’s theorem
  • Homomorphisms

Examples:

  • (ℤ, +) → abelian group
  • (ℝ{0}, ×) → abelian group
  • S₃ → non-abelian
  • GL(n,ℝ) → non-abelian