Quotient group

Definition

Given a normal subgroup $N \triangleleft G$ , the quotient group $G/N$ is the set of cosets $\{gN: g \in N\}$ with the operation $g_1 N . g_2 N = (g_1.g_2) N$ . We say $G$ is an extension of $G/N$ by $N$ .

We can conceptualise this as squashing the set $H$ down into the identity.

Properties

Basic

The quotient of a group with its trivial subgroup is isomorphic - essentially equal - to the original group.

Interesting

The first isomorphism theorem: given a homomorphism $\phi: G \to G'$ (where $G'$ is some other group), whose kernel is $\ker(\phi)=H$ , we have an isomorphism $G/\ker(\phi) \to \phi(G).$

Examples

Given any homomorphism $\phi: G_1 \to G_2$ , the kernel $\ker(\phi)$ is a normal subgroup of $G_1$ .