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:

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.