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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 ..

1 Answer

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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.

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