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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
asked in Graph Theory by Veteran (35.4k points)   | 299 views
which set ?

3 Answers

+4 votes
Best answer

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.

answered by Veteran (29.2k points)  
selected by
+1 vote
B abelian group
answered by (125 points)  
0 votes

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.

answered by Loyal (4.1k points)  
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