Proving G is Abelian from (a*b)^2=(a^2)*(b^2)

[fk378]

Homework Statement

If G is a group such that (a*b)^2=(a^2)*(b^2) for all a,b in G, show that G must be abelian.

The Attempt at a Solution

First, I tried to expand the binomial (a*b)^2 and set it equal to (a^2)*(b^2). But then I didn't know where to go from there.

 

[Dick]

You aren't trying very hard. A group is abelian if ab=ba for all a and b. (ab)^2=abab. (a^2)*(b^2)=aabb. Take it from there.