# quotient group

- also called factor group

## 1. definition from equivalence relation

Consider a group \(G\). For a quotient group, you have an equivalence relation. Everything in the same equivalence class is identified with elements of the same class, i.e., you consider them to be inter-changeable.

Consider elements \(a\) and \(b\) from the group, who belong to different equivalence classes. The group operation on \(a\) and \(b\), produces an item that belongs to the same equivalence class that \(ab\) does

## 2. definition from normal subgroup

The equivalence class that contains the identity element is always a normal subgroup \(N\). It turns out that the other equivalence classes are the cosets of \(N\). (For a normal subgroup, the left and right cosets are the same)

Then, these cosets form a group. That is, consider the set of cosets \(G / N = \{gN | g \in G\}\) (remember there may be fewer than \(|G|\) cosets in the resulting set)

Define the following group operation on this set: for \(aN, gN \in G / N\), the result of \((aN)(gN)\) is \((ag)N\), i.e., the equivalence class that contains \(ag\).