Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. Therefore, a natural question is as follows:
Fix $n\in\mathbb{N}$. Suppose that $G_n$ is the set of all groups of order $n$ and $L_n$ is the set of all Latin squares of order $n$. We shall then define the relation $R\subset G_n \times L_n$ by $(g,\ell)\in R$ iff there exists a labeling of the elements of $g$ such that the Cayley table of $g$, ignoring the headings, is $\ell$, for $g\in G_n$ and $\ell\in L_n$. What are the properties of $R$?
It's not too hard to see that every group $g$ of order $n$ is related to $n!$ Latin squares, because permuting the elements of $g$ in its Cayley table will produce different Latin squares.
Suppose that $RL_n$ is the set of all reduced Latin rectangles of order $n$, then we have that each $g\in G_n$ is related to a unique $\ell\in RL_n$. Thus, the relation $R'\subset G_n \times RL_n$, defined similarly to R as above, is in fact a mapping. It is clear that $R'$ is not an injective mapping, because if $g,h\in G_n$ are isomorphic, then $g$ and $h$ will both be related to the same $\ell \in RL_n$. Furthermore, $R'$ is not, for a general $n$, surjective. This is because some Latin squares are only related, in the "natural" fashion, to non-associative quasi-groups. So, yet another question naturally arises:
Suppose that im$(R')$ is the set of all $\ell\in RL_n$ such that $(g,\ell)\in R'$ for some $g\in G_n$. What are the properties of $\ell\in RL_n \cap im(R')^C?$ Well, obviously such an $\ell$ will be related to, in the "natural" fashion, to a non-associative quasi-group. However, what is it about the structure of the Latin square specifically that is not shared by the Cayley table of any group?