Question

if (G,*) is group,then show that (a^-1)^-1=a​

Answer 1

Note :

• Group : An algebraic system (G,*) is said to be a group if the following condition are satisfied :

1. G is closed under *
2. G is associative under *
3. G has a unique identity element
4. Every element of G has a unique inverse in G

• Abelian group : If a group (G,*) also holds commutative property , then it is called commutative group or abelian group , ie . if x*y = y*x ∀ x , y ∈ (G,*) , then the group G is said to be abelian .

Solution :

• Given : (G,*) is a group

• To prove : (a⁻¹)⁻¹ = a

Proof :

Let a ∈ G be any arbitrary element and e be the identity element in G .

Now ,

→ (a⁻¹)⁻¹ = (a⁻¹)⁻¹*e

→ (a⁻¹)⁻¹ = (a⁻¹)⁻¹*(a⁻¹*a) [°•° a⁻¹*a = e]

→ (a⁻¹)⁻¹ = [(a⁻¹)⁻¹*a⁻¹]*a [°•° associativity]

→ (a⁻¹)⁻¹ = e*a [°•° (a⁻¹)⁻¹*a⁻¹ = e]

→ (a⁻¹)⁻¹ = e [°•° e*a = a]

Hence proved .