let G be group such that (a*b)^2 = a^2 * b^2 for a,b element G, show that is abelian.

Question

let G be  group such that (a*b)^2  = a^2 * b^2 for a,b element G,
show that <G,*> is abelian.

anonymous · Answer

You know that
$$(G,*)\text{ is a group. To show that it's abelian, you must show commutativity,}\
\text{i.e. }\;\;a*b=b*a$$

anonymous · Answer

d you mean that i must say
(a * b)^2 = a^2 * b^2 = b^2 * a^2 = ( b * a)^2
thus <G , * is abeian

anonymous · Answer

Is the operation you're using here multiplication? Or just the general "star"? If it's the second, I don't think you can make that conclusion right away.

anonymous · Answer

the oparation is just  general star

anonymous · Answer

You seem to have treated * as multiplication, which doesn't work. For this particular operation, I'm not sure how to show whether it's commutative in this case. It's the (a*b)^2 that's bothering me.