Cyclic group/Canonical representation/Example

The subgroups of the integers are of the form ${}\mathbb {Z} n$ with ${}n\geq 0$, due to fact. The factor group are denoted by

$\mathbb {Z} /(n)$

("${}\mathbb {Z}$ modulo ${}n$“). For ${}n=0$, this is just ${}\mathbb {Z}$ itself; for ${}n=1$, this is the trivial group. In general, the equivalence relation on ${}\mathbb {Z}$ defined by the subgroup ${}\mathbb {Z} n$ is given in the way that ${}a$ and ${}b$ are equivalent if and only if their difference ${}a-b$ belongs to ${}\mathbb {Z} n$, that is, if it is a multiple of ${}n$. Therefore, (${}n\geq 1$), every integer number is equivalent to exactly one of the ${}n$ numbers

$0,1,2,\ldots ,n-1$

(or, as we also say, congruent modulo ${}n$), namely to the remainder upon division through ${}n$. These remainders form a system of representatives for the factor group, and contains ${}n$ elements. The fact that the quotient mapping

$\mathbb {Z} \longrightarrow \mathbb {Z} /(n),a\longmapsto [a]=a\!\!\!\mod n,$

is a homomorphism might be expressed by saying that the remainder of a sum of two integers depends only on their remainders, not on the numbers themselves. As an image of the cyclic group ${}\mathbb {Z}$, the group ${}\mathbb {Z} /(n)$ is also cyclic; ${}1$ (but also ${}-1$) is always a generator.