Modular Arithmetic in the Integers

Definition: Start with $\mathbb{Z}$. $\mathbb{Z}_{11}$ are the equivalence classes for the equivalence relation:
$a \equiv_{11} b$ $iff$ $a-b=m*11$.

$\mathbb{Z}_{11}$ is a group.

Subgroups of $\mathbb{Z}_{11}$: Only $\mathbb{Z}_{11}$ itself is a subgroup.

Generators: $\mathbb{Z}_{11} = <1>$, $<2>$, $<3>$, $<4>$, $<5>$, $<6>$, $<7>$, $<8>$, $<9>$, $<10>$.

Cosets of $\mathbb{Z}_{11}$: $\mathbb{Z}_{11}$ is the only coset since the only subgroup is $\mathbb{Z}_{11}$.