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).