Same order of element in group... If we have two elements $a$ and $b$ in a group such that $a^3 = e$ and $b^3 = e$ does this imply that $a=b$? I suspect so, but I think im missing something trivial here?
No: Another counterexample: 

Take $(1, 2, 3),\;(1, 3, 2) \in S_3$. Each of these permutations is of order $3$ (confirm that, if you need to), yet clearly $(1, 2, 3) \neq (1, 3, 2)$.
No. Consider the group of integers mod 3.