Question

Q.5. Theorem : Reversal Rule
Prove that the inverse of the product of two elements of group is the
product of the inverses of the elements in the reverse order.​

Answer 1

Step-by-step explanation:

thanks for free point s

Answer 2

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

• Moreover , if a group (G,*) also holds commutative property , then it is called commutative group or abelian group .

Solution :

To prove :

In a group G , (ab)⁻¹ = b⁻¹a⁻¹ ∀ a , b ∈ G .

Proof :

Let a , b € G and let a⁻¹ and b⁻¹ be the inverse elements of a and b respectively , then

a⁻¹a = aa⁻¹ = e and b⁻¹b = bb⁻¹ = e where e is the identity element in G .

Now ,

(ab)(b⁻¹a⁻¹) = [a(bb⁻¹)]a⁻¹

= (ae)a⁻¹

= aa⁻¹

= e

Also ,

(b⁻¹a⁻¹)(ab) = b⁻¹[(aa⁻¹)b]

= b⁻¹(eb)

= b⁻¹b

= e

Thus , (ab)(b⁻¹a⁻¹) = (b⁻¹a⁻¹)(ab) = e

→ (ab)⁻¹ = b⁻¹a⁻¹ ∀ a , b ∈ G

Hence proved .