Prove this is true for every group G, or give a counter example to show this is false in some groups:
If x^2 = a^2, then x=a
Obviously, you need a counterexample to $\displaystyle x^2=a^2\Rightarrow x=a$. - What about $\displaystyle \mathbb{R}\backslash\{0\}$ with multiplication?
Another, simpler, example is the set {1, -1} with multiplication as operation. Yet, another, the Klein 4-group. In fact, any finite group whose order is even!