Let G be a group and suppose that ab2 a2b2 for all a and b

Let G be a group and suppose that (ab)^2 = (a^2)(b^2) for all a and b in G. Prove that G is an abelian group.

Solution

For all a, b in G, (ab*)(ab) = aabb.

i.e. abab = aabb.

Cancel a on the LHS and b on the RHS

we get ba = ab.

A group G is said to be abelian if ab = ba for all elements a,b in G

Hence g is an Abelian.