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If $(G , .)$ is a group such that $(ab)^{-1}=a^{-1}b^{-1},\forall a,b \in G,$ then $G$ is a/an

1. Commutative semi group
2. Abelian group
3. Non-abelian group
4. None of these
which set ?

In a group (G , .) is said to be abelian if

(a*b) =(b*a) ∀a,b ∈G

(ab)-1 = (b--1a-1)............(1)

Given ,(ab)-1= a-1b-1     ..........(2)

from (1) and (2) we can write

a-1b-1   b-1 a-1

we can also write it as

ab=ba

Hence,Option(B)Abelian Group is the correct choice.

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+1 vote
B abelian group

A group is called as abelian group if and only if it fallow commutative rule properly.

Example as::

a*b=b*a

Now According to given problem

let:

(x*y)^1=(y*x)^-1

x^-1y^-=y^-1x^-1

so xy=yx that's why it abelian group.