I am currently trying to teach myself abstract algebra by working through Dummit and Foote's text, and one of the exercises asks us to prove that the group table for a specific group $G = \{1, a, b, c\}$ with all elements of order $\leq 3$ is unique. Does this simply mean that every entry of the group table has a unique value? This seems obvious for any group $G$, since for any $w,x,y,z\in G$, if $w \neq z$ but $xy = w$ and $xy = z$, then $y = x^{-1}w$ and $y = x^{-1}z$, which leads to the contradiction $w=z$ via the cancellation law.
I just wanted to make sure that I had the right definition in mind, since the other question on here about this exercise does not directly address this part of the exercise. They simply determine what the value of each product must be explicitly in the course of proving that the specific $G$ in the exercise is abelian.
Proving a group $G$ has a unique group table means that any table you create for the group can have its rows/columns rearranged to look like just one table.
I recommend writing out every table for groups of order $4$ first (you should get four different tables). At that point look carefully and you'll realize three of the tables are technically "the same" if you reorder the rows/columns (you'll learn later this means they're isomorphic). So then you'll have shown all groups with four elements only correspond to two different tables. What's left is to look at the order of the elements to answer your question.