| A group is a set of elements G = {a, b, c, ...},
together with an operation, usually denoted + or .,
that takes pairs of elements of G and returns elements of G. |
| We say G is closed under the operation. |
| The operation is associative: a+(b+c) = (a+b)+c
for all elements a, b, and c of G. |
| The group G has an
identity element e having the property that
e + a = a + e = a for all a in G. |
| Finally, each a in G has an inverse,
b in G having the property that a + b = b + a = e. |