Question

Let G be a group. Prove that G is an abelian group if and only if (ab)^2 = (a^2)(b^2) for every a, b element of G

Answer 1

@satellite73

Answer 2

again one way is easy
if it is abelian then \((ab)^2=abab=aabb=a^2b^2\)

Answer 3

abelian allowed you to write \(abab\) as \(aabb\) because in the center \(ba=ab\)

Answer 4

now suppose for all \(a\) and \(b\) you have \((ab)^2=a^2b^2\)
that means for all \(a, b\) you have \(abab=aabb\)

Answer 5

take the equation
\[abab=aabb\] multiply on the left by \(a^{-1}\) and on the right by \(b^{-1}\) and you get
\[ba=ab\] which is what you wanted to show