\chapter{Group homomorphisms}


\section{Group homomorphism}

A powerful tool in abstract algebra is the idea of a \emph{homomorphism}. Sometimes different groups can have similar behaviours, and homomorphisms are the primary technique by which this is expressed.



\begin{definition}[Group homomorphism]
Let $(G,\circ)$ and $(H,*)$ be two groups. A \emph{group homomorphism} $f : G \to H $ is a function between groups that 'preserves' the group's operation in the following sense; if $g_1,g_2$ are elements of $G$, we have the following.
\[ f(g_1 \circ g_2) = f(g_1) * f(g_2) \]
\end{definition}

Homomorphisms have some elementary properties relating to mappings of identity and inverse elements.

\begin{proposition}
\[ f(1_{G})= 1_{H}\]
\end{proposition}

\begin{proposition}
\[ f(g)^{-1}= f(g^{-1})\]
\end{proposition}




\begin{definition}[Group monomorphism]
injective homomorphism
\end{definition}
\begin{definition}[Group epimorphism]
surjective homomorphism
\end{definition}
\begin{definition}[Group endomorphism]
\end{definition}

I've listed terminology for injective and surjective homomorphisms, however I will defer 'bijective homomorphisms' for their own section since they are so special.


\subsection{Examples of Group homomorphism}


\begin{example}[The determinant]
	\[\mathrm{det} : \mathrm{GL}(n,\mathbb{R}) \to (\mathbb{R} \setminus \{0\},\times) \]
\end{example}

We know from linear algebra that $\mathrm{det}(\mathbf{A} \mathbf{B}) = \mathrm{det}(\mathbf{A}) \mathrm{det}(\mathbf{B})$ and that $\mathrm{det}(\mathbf{I})=1$; these are the exact conditions of a homomorphism.





\begin{example}[Permutation signature]
\end{example}


When we learn about special group constructions (quotient groups, product groups, etc.) we will rely heavily on the notions of homomorphisms to describe the behaviour of the constructions we create. When we have these tools available to us, we can in turn find more interesting homomorphisms lurking between related constructs.


These types of group homorphisms have particularly interesting properties, but group isomorphisms are perhaps the most important class of homomorphisms.

\section{Group isomorphism}


One of the most notable types of homomorphism is the \emph{isomorphism}. This mapping expresses groups that have \emph{exactly} the same behavior. To express this, we want our homomorphism to be bijective, since this means that every element in the domain group has a direct counterpart in the codomain group and vice versa.

\begin{definition}[Group isomorphism]
Let $(G,\circ)$ and $(H,*)$ be two groups. A \emph{group isomorphism} $f : G \to H $ is a bijective isomorphism.
If there exists an isomorphism between $G$ and $H$, then the groups are \emph{isomorphic} to eachother, also written as $(G,\circ) \cong (H,*)$ or if the group operations are known, $G \cong H$.
\[G \text{ is isomorphic to } H \iff \exists f : G \to H [ f \text{ is a homomorphism } \land f \text{ is bijective}] \]
\[G \cong H \iff G \text{ is isomorphic to } H \]
\end{definition}

When groups are isomorphic, mathematicians tend to view them as the same group 'up to a change of symbols'; isomorphic groups have the exact same behaviour as a group, and the isomorphism is used to translate which symbol in the domain group acts identically to which symbol in the codomain group. They'll often say such groups are equal \emph{up to an isomorphism}.


It's important to note that although as groups isomorphic groups behave identically, when considering external spaces and structures the groups may exhibit different behaviour; the isomorphic relation is not equality.



\begin{definition}[Group automorphism]
Let $(G,\circ)$ be a group. A \emph{group automorphism} $f : G \to G $ is an isomorphism onto the same group.
\end{definition}

\begin{proposition}
For any group $(G,\circ)$ there exists a class of automorphisms called the \emph{inner automorphisms} defined as $\varphi_g (x) = gxg^{-1}$.
\end{proposition}


\subsection{Properties of Group isomorphism}

According to the way I have described the desired properties of the isomorphism, we would hope that the isomorphism relation is an equivalence relation, which indeed it is.


\begin{proposition}
The group isomorphism relation $\cong$ is an equivalence relation.
\end{proposition}


\subsection{Kernel of a group homomorphism}



%Some Theorems

A \emph{Canonical map} is a function between two objects that arises from their definitions. It is a function used to define the behaviour of some object.


From set theory we are familiar that functions give rise to the ideas of domains, codomains, images, preimages, fibers etc. however when considering group homomorphisms we can now define the idea of a \emph{kernel}; the set of all elements in a homomorphism mapped to the identity.


\begin{definition}[Kernel (group homomorphism)]
Let $(G,\circ)$ and $(H,*)$ be two groups and $f : G \to H$ a homorphism. The \emph{kernel of a group homomorphism} is the set of elements that a homomorphism maps to the other group's identity element.
\[ \mathrm{ker}(f) = \{ g \in G: f(g)=1_{H} \} \]
\end{definition}

\begin{proposition}
\[ f : G \to H \text{ is a group homomorphism } \implies \mathrm{ker}(f) \leq G \]
\end{proposition}


\begin{proposition}
	\[ f : G \to H \text{ is a group isomorphism } \implies \mathrm{ker}(f) =\{1_G\} \]
\end{proposition}




\subsection{Examples of isomorphisms}

\begin{example}[The exponential function]
\[\mathrm{exp} : (\mathbb{R},+) \to (\mathbb{R}_{+},\times) \]
\[\mathrm{exp}(x)=e^x\]
\end{example}

The fact that for exponentials, addition in the exponent is multiplication of factors, is really a homomorphism between addition over the reals to multiplication over positive reals (since exponentials are always greater than 0 with real exponents). This function is particularly interesting since it is also a bijection; bijective homomorphisms are called \emph{isomorphisms}, and we will have much to say about them.





Groups can be used to prove a key insight in about metric spaces; isometries are simply reflections and rotations. In terms of isomorphisms, this is saying that the group of isometries on a space $X$ is isomorphic to the group $\mathrm{O}(n,X)$.



