Show that the group G is abelian if (ab)^2=a^2b^2

Modern Algebra
Group Theory (VIII)

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

Or, Show that the group G is abelian iff (ab)^2=a^2b^2.

The fully formatted problem is in the attached file.

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Solution Summary

This shows how to complete a proof regarding an Abelian group. The solution is detailed and well presented.