GATE CSE
First time here? Checkout the FAQ!
x
0 votes
141 views

Question


IF ( G, * ) is an abelian group then discuss correctness of each of the options ...

a)  a = a-1 for all a $\epsilon$ G

b) a2 = a for all a $\epsilon$ G

c) (a*b)2 = a2 * b2 for all a,b $\epsilon$ G

d) G is finite order

My Approach

Abelian Group -- > Commutative Group

A) 

Let  (a*b)-1= b-1 * a-1

    givent that a-1 = a

   Therefore , (a*b)-1 = b * a

  Therefore , G is a abelian group   Given statement is correct

B)

a2 = a 
    IF a*a = a THEN a is identity element (e)

 

C)

( a*b)= a2 * b2

 (a*b)*(a*b) = (a*a) * (b*b)

a* (b*a)*b = a*(a*b)*b

  b*a = a*b   // cancelling a using left cancellation law and b using right cancellation law

  Therefore G is abelian group Given statement is correct

D)


PLEASE HELP ME ..

asked in Set Theory & Algebra by Veteran (20.2k points)   | 141 views
what are you proving here?
@arjun sir , chechking the correctness of statements given
But you are asked to prove statement 1. Now, you assumed statement 1 true for A and took another assumption and proved G is abelian. It should be done like this: G is abelian, use properties of abelian group and show that statement A is true. Or just give a counter example showing statement A is false.
Ok. sir

2 Answers

0 votes
D. We have to prove that every |G| is finite for abelian groups.

Since all abelian groups are cyclic i.e a^n = a; for some n.

now consider n is infinite, which implies this is not a cyclic group, hence not an abelian. Therefore n is finite.

Thus |G| = n which is finite.
answered by Active (1.1k points)  
–1 vote
Youb r right abolutely,then why r u asking ?
answered by Loyal (4.5k points)  


Top Users Mar 2017
  1. rude

    5236 Points

  2. sh!va

    3054 Points

  3. Rahul Jain25

    2920 Points

  4. Kapil

    2732 Points

  5. Debashish Deka

    2602 Points

  6. 2018

    1574 Points

  7. Vignesh Sekar

    1430 Points

  8. Bikram

    1424 Points

  9. Akriti sood

    1420 Points

  10. Sanjay Sharma

    1128 Points

Monthly Topper: Rs. 500 gift card

21,549 questions
26,889 answers
61,249 comments
23,251 users