![]() Abelian Groups An arithmetic operation is said to be commutative if the
order of its arguments is insignificant. With ordinary numbers, addition
and multiplication are commutative operations; for example, 2* 9 = 9*2 and
2 + 9 = 9 + 2. However, subtraction and division are not commutative since
2 - 9 <> 9 - 2 and 2 / 9 <> 9 / 2.
A group is called abelian if its main operation is commutative. Thus an additive group is abelian if a + b = b + a for all elements a, b in the group. A multiplicative group is abelian if a b = b a for all elements a, b in the group. The additive group Zn and the multiplicative group Zp* are both abelian groups. |