# Quotient groups

The first step in this direction is to define cosets. For any subgroup $${H\subset G}$$ and element $${g\in G}$$, the set $${gH\equiv\left\{ gh\mid h\in H\right\} }$$ is called a left coset of $${H}$$ in $${G}$$. Right cosets are defined similarly, and $${g}$$ is called a representative element of the coset $${gH}$$. Cosets can be viewed as a partitioning of all of $${G}$$ into equal-sized disjoint “copies.” However, this cannot be used to define a group quotient $${G/H}$$ since in general, the cosets themselves do not form a group.

A normal subgroup (AKA invariant subgroup, self-conjugate subgroup) of $${G}$$, denoted $${N\triangleleft G}$$, is defined as follows:

$$\displaystyle N\triangleleft G\:\textrm{if}\: gN=Ng\:\forall g\in G\Leftrightarrow gNg^{-1}\subseteq N\:\forall g\in G$$

Note that an immediate consequence of the above definition is that any subgroup of an abelian group is normal. It is not hard to see that the cosets of $${N}$$ in $${G}$$ (left and right being identical) do in fact comprise the elements of a group under the group operation $${(gN)(hN)\equiv\left\{ gnhm\mid n,m\in N\right\} =(gh)N}$$. We denote this group $${G/N}$$ and call it a quotient group (AKA factor group).

The kernel of a homomorphism $${\phi}$$ from $${G}$$ to another group $${Q}$$ is the subgroup of elements that are mapped to the identity $${\mathbf{1}}$$. Any normal subgroup $${N}$$ is then the kernel of the group homomorphism $${\phi\colon G\to G/N}$$ defined by $${g\mapsto gN}$$, and thus all normal subgroups are homomorphism kernels. The converse of this is also true: for any homomorphism $${\phi\colon G\to Q}$$, $${\textrm{Ker}\phi}$$ is normal in $${G}$$. Furthermore, the quotient group is isomorphic to the subgroup $${\phi\left(G\right)}$$ of $${Q}$$, so that we have the equation $${G/\textrm{Ker}\phi\cong\phi\left(G\right)}$$, called the first isomorphism theorem or the fundamental theorem on homomorphisms: $${\phi}$$ shrinks each equal-sized coset of $${G}$$ to an element of $${\phi\left(G\right)}$$, which is therefore a kind of simpler approximation to $${G}$$.

It is helpful to demonstrate quotient groups with an easy example. Let $${\phi\colon\mathbb{Z}\to\mathbb{Z}_{3}}$$ be the (surjective) homomorphism that sends each element to its remainder after being divided by 3. The kernel of this homomorphism is the subgroup of $${\mathbb{Z}}$$ consisting of all integers of the form $${3n}$$. Then the cosets $${\mathbb{Z}/\left\{ 3n\right\} }$$ are the subgroups $${\left\{ 3n\right\} }$$, $${\left\{ 3n+1\right\} }$$, and $${\left\{ 3n+2\right\} }$$, which are isomorphic to the elements of $${\mathbb{Z}_{3}}$$.