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)2 = 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 ..