Showing that for some group it is abelian iff $x • (y • x ^{−1} ) = y$

Question

I have to show that for some group it is abelian iff $x • (y • x^{−1}) = y$. This is what I did: Starting with the given statement $x • (y • x^{−1}) = y$ which implies $x • (y • x^{−1}) • x= y • x$. Since it is a group associativity can be used giving $(x • y) • (x^{−1} • x) = y • x$ implying $x • y = y • x$. So since the group commutes it can be implied to be abelian. Is this sufficient? Thank you

Answer 1

No, it is not. All you did was to prove that if that condition holds, then the group is Abelian. But you also have to prove it in the other direction. That is not difficult, though: if the group is Abelian, then$$x\bullet(y\bullet x^{-1})=x\bullet(x^{-1}\bullet y)=(x\bullet x^{-1})\bullet y=y.$$

Answer 2

This trivially follows from the following:

$$\begin{align} x \bullet (y \bullet x^{-1}) = y &\iff (x \bullet y) \bullet x^{-1}= y \\ &\iff ((x \bullet y) \bullet x^{-1})\bullet x =y \bullet x\\ &\iff (x \bullet y) \bullet (x^{-1} \bullet x) = y \bullet x\\ &\iff (x \bullet y) \bullet 1 = y \bullet x\\ &\iff x \bullet y = y \bullet x. \end{align}$$

Here I denoted the neutral element of the group with $1$. Try to understand why each equivalence holds (good exercise!).