Loop not implies group

Statement

It is possible to have an algebra loop that is not a group.

Proof

Consider the algebra loop with multiplication table:

$*$	1	2	3	4	5
1	1	2	3	4	5
2	2	1	5	3	4
3	3	5	4	2	1
4	4	3	1	5	2
5	5	4	2	1	3

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, $3*(3*3)\neq (3*3)*3$).