Cosets and Equivalence Relations

Question

Today in math my professor was talking about equivalence relations and modulo. Then he said something about cosets, and I don't understand the relation. I read that a coset was a subgroup with all if it's elements added with another element of the group. And I guess equivalence relations for subgroups, at least the integers mod some number? My main question is how are cosets and equivalence relations related?

Answer 1

Cosets and equivalences relations are not related, except inasmuch as you can define an equivalence relation fairly naturally on the cosets of some subgroup of a group.

For example, taking the group $G = (\mathbb{Z}, +)$, and its subgroup $(3\mathbb{Z},+)$ gives you an infinite number of cosets $(\dots,-1 + 3\mathbb{Z}, 0 + 3\mathbb{Z}, 1 + 3\mathbb{Z}, 2 + 3\mathbb{Z}, 4 + 3\mathbb{Z},\dots)$. However, as $1 + 3\mathbb{Z}$ and $4 + 3\mathbb{Z}$ define the same sets, they are equivalent, both being the set of numbers equivalent to $1$ mod $3$.