Loop not implies group

Statement
It is possible to have a fact about::loop that is not a fact about::group.

Proof
Consider the algebra loop with multiplication table:

In other words, this is the algebra loop corresponding to the Latin square:

$$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 3 & 4 \\ 3 & 5 & 4 & 2 & 1 \\ 4 & 3 & 1 & 5 & 2 \\ 5 & 4 & 2 & 1 & 3 \\\end{pmatrix}$$

This is clearly not a group (for instance, $$3 * (3 * 3) \ne (3 * 3) * 3$$).