The group Zn
The group Zn uses only the integers from 0 to n - 1. Its basic operation is addition, which ends by reducing the result modulo n; that is, taking the integer remainder when the result is divided by n. One very important feature of arithmetic in a group is that all calculations give numbers which are in the group; this is called closure. Modular reduction by n ensures that all additions result in numbers between 0 and n - 1.
The additive group Z15 uses the integers from 0 to 14. Here are some sample additions in Z15:
10 + 12 mod 15
= 22 mod 15
= 7
4 + 11 mod 15
= 15 mod 15
= 0.
In Z15, 10 + 12 = 7 and 4 + 11 = 0. Notice that both calculations have answers between 0 and 14.
Additive Inverses
Each number x in an additive group has an additive inverse element in the group; that is an integer -x such that x + (-x) = 0 in the group. In Z15, -4 = 11 since 4 + 11 mod 15 = 15 mod 15 = 0.
Other operations
While addition is the main operation in the additive group Zn, other operations can be derived from addition. For example, the subtraction x - y can be performed as the addition x + (-y) mod n. In Z15, 1 - 4 = 1 + (-4) = 1 + 11 mod 15 = 12.
It is also possible to define multiplication in Zn by repeated addition. For example, the multiplication 4 9 in Z15 can be achieved by adding together 9 + 9 + 9 + 9 mod 15 = 36 mod 15 = 6.
