A
binary operation on a
set (call it
A) containing an
identity element where:
For every a e A and b e A,
there exists a unique u e A such that au = b, and
a unique v e A such that va = b.
The primary difference between a loop and a group is that a loop is
not necessarily associative, where a group is associative (in fact a
group is the same thing as an associative loop).