This is more of a theoretical question, but I'm having trouble understanding why it is that Latin squares are generalizations of a group?
I kind of arrived at this question trying to figure out why independent vectors were so important in linear algebra.
To phrase the question differently, why are groups composed of a set n with n objects and n permutations?
To put together the two comments by Matt Samuel and jbuddenh, we can certainly claim that
Every latin square is the multiplication table of a quasigroup. A quasigroup is a magma (groupoid) $G$ where, for all $a\in G$, the left and right translation $L_a(x)$, $R_a(x)$ are bijections of $G$ (permutations). This is, a quasigroup is a magma (groupoid) $G$ with left and right cancellation and left/right division (see definitions 1 to 12 in this paper).
Among all quasigroups, only those showing associativity are groups. Hence, in front of a multiplication table that is a latin square, one must test associativity before claiming that is a group.
For example,
$$\begin{array}{c|ccccc} \circ & 1 & 2 & 3 & 4 & 5\\ \hline 1 & 1 & 2 & 3 & 4 & 5\\ 2 & 2 & 3 & 4 & 5 & 1\\ 3 & 3 & 4 & 5 & 1 & 2\\ 4 & 4 & 5 & 1 & 2 & 3\\ 5 & 5 & 1 & 2 & 3 & 4\\ \end{array}$$
is a group while
$$\begin{array}{c|ccccc} \bullet & 1 & 2 & 3 & 4 & 5\\ \hline 1 & 1 & 2 & 3 & 4 & 5\\ 2 & 2 & 1 & 4 & 5 & 3\\ 3 & 3 & 5 & 1 & 2 & 4\\ 4 & 4 & 3 & 5 & 1 & 2\\ 5 & 5 & 4 & 3 & 2 & 1\\ \end{array}$$
is only a quasigroup (with identity, $1$), although
$$\begin{array}{ccc} \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1\\ 3 & 4 & 5 & 1 & 2\\ 4 & 5 & 1 & 2 & 3\\ 5 & 1 & 2 & 3 & 4\\ \end{array} & \text{ and } & \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 2 & 1 & 4 & 5 & 3\\ 3 & 5 & 1 & 2 & 4\\ 4 & 3 & 5 & 1 & 2\\ 5 & 4 & 3 & 2 & 1\\ \end{array} \\ \end{array}$$
are both latin squares (reference: page 5 of this book).
It is often thaught that a set together with a binary operation is called a magma or a groupoid, an associative groupoid is a semigroup, a semigroup with identity is a monoid and, finally, a monoid with inverses is a group. On the other hand, a groupoid with left/right cancellation and division is called a quasigroup and an associative quasigroup is a group (an associative quasigroup also have identity).