a quotient group is a group which is the product of mapping things out.

## subgroups

The set of integers \(\mathbb{Z}\) is obviously a group. You can show it to yourself that multiples of any number in the group is a subgroup of that group.

For instance:

\(3 \mathbb{Z}\), the set \(\{\dots -6, -3, 0, 3, 6, \dots\}\) is a subgroup

## actual quotient groups

We can use the subgroup above to mask out a group. The resulting product is **NOT** a subgroup, but its a new group with individual elements being subsets of our original group.

For instance, the \(\mod 3\) quotient group is written as:

\begin{equation} \mathbb{Z} / 3 \mathbb{Z} \end{equation}

Each element in this new group is a set; for instance, in \(\mathbb{Z} / 3\mathbb{Z}\), \(0\) is actually the set \(\{\dots -6, -3, 0, 3, 6, \dots\}\) (i.e. the subgroup that we were masking by). Other elements in the quotient space (“1”, a.k.a. \(\{ \dots, -2, 1, 4, 7 \dots \}\), or “2”, a.k.a. \(\{\dots, -1, 2, 5, 8 \dots \}\)) are called “cosets” of \(3 \mathbb{Z}\). You will notice they are not a subgroups.