## Group

A non-empty set G of some elements (a, b, c, etc.), with one or more operations is known as a **group.**

## A set needed to be satisfied following properties to become a group:

**1) Closure Property:**

a.b ∈ G , ∀ a, b ∈ G

**2) Associative Property:**

(a . b) . c = a . (b . c), ∀ a, b, c ∈ G

**3) Existence of Identity:**

e → identity element

e.a = a = a.e, ∀ a ∈ G

**4) Existence of Inverse:**

a-1→ inverse of a

a.a-1 = e = a-1.a , ∀ a ∈ G

## Abelian or Commutative Group

### A set needed to be satisfied following properties to become an abelian group:

**1) Closure Property:**

a.b ∈ G , ∀ a, b ∈ G

**2) Associative Property:**

(a . b) . c = a . (b . c), ∀ a, b, c ∈ G

**3) Existence of Identity:**

e → identity element

e.a = a = a.e, ∀ a ∈ G

**4) Existence of Inverse:**

a-1 → inverse of a

a.a-1 = e = a-1.a , ∀ a ∈ G

**5) Commutativity:**

a.b = b.a , ∀ a , b ∈ G

## Subgroup

A subgroup is a subset H of group elements of a group G that satisfies all the four properties of a group.

“ H is a subgroup of G” can be written as H ⊆ G

A subgroup H of a group G, where H ≠ G, is known as proper subgroup of G.