![]() The group Zn The group Znuses 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. |