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) a^{2} = a for all a $\epsilon$ G
c) (a*b)^{2} = a^{2} * b^{2} 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)
a^{2} = a
IF a*a = a THEN a is identity element (e)
C)
( a*b)^{2 }= a^{2} * b^{2}
(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 ..