Homomorphisms

Definition and Examples

Definition: A homomorphism $\phi:G\to H$ is a function that satisfies the property $\phi(g_1g_2)=\phi(g_1)\phi(g_2)$ for all $g_1,g_2\in G$. An isomorphism is a homomorphism that is bijective.

A homomorphism is a map that gives a partial relation between the structure of $G$ and $H$.

Examples:

Let $G=S_{n}$ and $H=\mathbb{Z}_{2}$. Define $\phi(\sigma) = \begin{cases} 0 & \hbox{if $\sigma$ is an even permutation} \\ 1 & \hbox{if $\sigma$ is an odd permutation} \end{cases}$ Then $\phi$ is a homomorphism.
Let $G=\mathrm{GL}_{2}(\mathbb{R})$ and let $H=\mathbb{R}^{\times}$. Then $\phi(g)=\mathrm{det}(g)$ is a homomorphism.
Let $H$ be any group and let $h\in H$ be an element. Define $\phi:\mathbb{Z}\to H$ by $\phi(n)=h^{n}$. Then $\phi$ is a homomorphism.
Let $G=\mathbb{R}$ and $H=\mathbb{T}$, the group of complex numbers of norm $1$ with multiplication. Then the map $\phi(r)=\mathrm{cis}(r)=\cos(r)+i\sin(r)$ is a homomorphism.

Key Properties of Homomorphisms

See this video and these notes.

Proposition: (Proposition 11.4 of the text) Let $\phi: G_1\to G_2$ be a homomorphism. Then:

If $e_1$ is the identity of $G_1$, $\phi(e_1)$ is the identity of $G_2$.
If $g_1$ is an element of $G_1$, then $\phi(g_1^{-1})$ is the inverse $\phi(g_1)^{-1}$ of $\phi(g_1)$.
If $H_1$ is a subgroup of $G_1$, then the image $\phi(H_1)$ is a subgroup of $G_2$.
If $H_2$ is a subgroup of $G_2$, then the preimage $\phi^{-1}(H_2)$ is a subgroup of $G_1$.
If $H_2$ is a normal subgroup of $G_2$, then the preimage $\phi^{-1}(H_2)$ is a normal subgroup of $G_1$.

Kernels

See this video and these notes

Definition: The kernel of a homomorphism $\phi:G\to H$ is the preimage of the identity of $H$: $\mathrm{Ker}(\phi)=\phi^{-1}(\{e_H\}).$

Proposition: (Theorem 11.5 of the text) The kernel of a homomorphism $\phi:G\to H$ is a normal subgroup of $G$.

The kernel of the determinant is $\mathrm{SL}_{2}(\mathbb{R})$.
The kernel of the map $\phi: \mathbb{Z}\to G$ given by $\phi(n)=g^{n}$ is either ${0}$, if $g$ has infinite order, or the subgroup $k\mathbb{Z}$ where $k$ is the order of $g$ in $G$.
The kernel of the map $\mathrm{cis}:\mathbb{R}\to\mathbb{T}$ are the integer multiples of $2\pi$ in $\mathbb{R}$.

Isomorphism Theorems

See this video and these notes.

Let $G$ be a subgroup and $H$ be a normal subgroup of $G$. Then the map $\phi: G\to G/H$ defined by $\phi(g)=gH$ is a homomorphism called the natural homomorphism or the canonical homomorphism.

Theorem: (First isomorphism theorem) Suppose $\psi:G\to H$ is a group homomorphism and $K$ is the kernel of $\psi$. Let $\phi:G\to G/H$ be the canonical homomorphism. Then there exists a unique isomorphism $\eta:G/K\to \psi(G)$ such that $\psi=\eta\phi$.

In words: every homomorphism $\psi$ from $G$ to $H$ is a composition of two steps:

first, you take the natural map from $G$ to the quotient $G/K$ where $K$ is the kernel of $\psi$;
then you have an isomorphism from $G/K$ to a subgroup of $H$.

Or, put another way, the image of a homorphism of a group $G$ is isomorphic to a quotient of $G$.

Additional isomorphism theorems

Theorem: (Second isomorphism theorem) Let $H$ be a subgroup of $G$, not necessarily normal in $G$, and $N$ a normal subgroup of $G$. Then $HN$ is a subgroup of $G$, $H\cap N$ is a normal subgroup of $H$, and there is an isomorphism $\phi$ $\phi: H/(H\cap N) \to HN/N.$

Theorem: (Correspondence Theorem) Let $N$ be a normal subgroup of $G$. Then $H\mapsto H/N$ is a bijection between the set of subgroups of $G$ that contain $N$ and the set of subgroups of $G/N$. In this correspondence, normal subgroups correspond to normal subgroups.

Theorem: (Third isomorphism theorem) Let $G$ be a group and $N$ and $H$ be normal subgroups of $G$ with $N\subset H$. Then $G/H\stackrel{\sim}{=}\frac{G/N}{H/N}.$