GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
482 views

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 (36.3k points)   | 482 views
which set ?

3 Answers

+5 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 (29.5k 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 (4.5k points)  


Top Users Mar 2017
  1. rude

    4768 Points

  2. sh!va

    3054 Points

  3. Rahul Jain25

    2920 Points

  4. Kapil

    2734 Points

  5. Debashish Deka

    2592 Points

  6. 2018

    1544 Points

  7. Vignesh Sekar

    1422 Points

  8. Akriti sood

    1342 Points

  9. Bikram

    1312 Points

  10. Sanjay Sharma

    1126 Points

Monthly Topper: Rs. 500 gift card

21,508 questions
26,832 answers
61,091 comments
23,146 users