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Mathematics-Online course: Linear Algebra - Basic Structures - Groups and Fields

Residue Classes modulo n

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The set

$\displaystyle \{0,1,\ldots,n-1\} $

forms an Abelian group with respect to the addition modulo $ n$. This group is denoted by

$\displaystyle \mathbb{Z}_n = \mathbb{Z}\,\operatorname{mod}\,n\, . $

Note that the multiplication modulo $ n$ does not define a group-structure on $ \mathbb{Z}_n$ because 0 has no inverse element.
By means of example

$\displaystyle \mathbb{Z}_4 = \{0,1,2,3\} $

you can easily verify that addition and subtraction,resp. , are compatible with the modulo operation. For example, we have

$\displaystyle 1+3\,$mod$\displaystyle \,4 = 0,\quad 0-2\,$mod$\displaystyle \,4 = 2 $

etc.

But multiplication modulo $ 4$ does not lead to a group structure because we have

$\displaystyle 1 \cdot 2\,$mod$\displaystyle \,4 = 3 \cdot 2\,$mod$\displaystyle \,4 = 2\,. , $

This contradicts uniqueness of the neutral element.
  automatically generated 4/21/2005