A Question about element?

Question

if a group G is abelian and (a*b)^2 = a^2b^2 for all a,b belongs to G. Is it true that G is abelian iff every element is its own inverse? can you please explain why? thank you.

Answer 1

do you assume that G is abelian? or not?

Is it true that G is abelian iff every element is its own inverse? The answer is NO, there are plenty of abelian groups that do not have this property.
take Z_3, with addition: 1+1=2 mod 3.


now,
if you have a group where:
(a*b)^2 = a^2b^2 for every a and b,
then G has to be abelian:
proof: (ab)^2= abab=aabb
so we can multiply by the inverses of a and b
and get that ba=ab

Answer 2

No this is not true. Take for example, G to be the integers under addition. The inverse of n is then -n, which is usually not equal to n.