# Definition The quotient group (or factor group) is a [[group]] constructed with respect to a [[Normal Subgroup]] as the collection of its [[Coset|cosets]], with each [[Coset]] being considered an [[group element|element]] of the quotient group. It obeys the rules of groups: 1. *Multiplication:* (gNg)(hN) = ghN as shown in the theorem [[Normal Subgroup#^2f79f8|here]] for right cosets. 2. *Associativity:* holds since it holds for all the elements of the [[Coset|cosets]], by virtue of them being group elements. 3. *Identity:*: The identity element is $eN$, where $e$ is the coset with the identity element, and this is just $N$ itself. This is because multiplying any element in $N$ by the entire subgroup gives you $N$: $nN = N, n \in N$. 4. *Inverse:* $(g N)(g^{-1}N) = (N g)(g^{-1}N) = (N^2) = (eN)$ More rigorously, let $G$ be a [[group]] and $N$ a [[Normal Subgroup]]. Define the quotient group $G/N$ as the set of all left cosets of $N$ in $G$, i.e., $G/N = {gN|g \in G}$. Since we have $e \in N$, then $g \in gN$. # Properties * The quotient group $G/G$ is [[Group Isomorphism|isomorphic]] to the [[Trivial Group|trivial group]]. * $G/{e}$ is isomorphic to $G$ * |G : N| = |G/N| where |G : N| is the [[Subgroup Index|index]] of $N$ . # Fun Resources There is a fun discussion on quotient groups to be found [here](https://www.math3ma.com/blog/whats-a-quotient-group-really-part-1)