\chapter{Group-like Structures}


Regarding algebraic structures of 1 set and 1 operation, we study groups in greatest depth because many mathematical objects we know (operations on numbers, composition of functions, symmetries of a shape etc.) are representable by groups, and groups have a rich set of results. It is useful to know structures that are similar yet weaker to groups, for the sake of terminology or in the creation of more complex algebraic structures.

We skip the algebraic structure of a category; this is covered in Category Theory.

\section{Monoids}


Groups are very intersting structures to study, however sometimes what we want to model doesn't quite have the properties of a group. There are a range of group-like structures that relax some of the properties of a group. Perhaps the runner up to the group is the \emph{monoid}.

Monoids are essentialy groups without the guarantee of elements being invertible. One notable place where monoids arise is category theory, where a category of one object is a monoid.

Monoids also find wide usage in theoretical computer science, specifically in automata theory.

\section{Magmas}


\section{Loops}




\part{Advanced}



\chapter{Free groups}

Recall how we could generate subgroups from one group element. We will now study how to do this with an arbitrary subset of a group, and eventually, we will form a group from just an arbitrary set!

Let's find a way to generate the smallest subgroup containing some arbitrary subset $X$, rather than just an element. SInce we know that taking interesections preserves the criterion for a group, we can take the intersection of all subgroups containing $X$. In the end, we obtain the smallest subgroup containing $X$.

\begin{definition}[Subgroup generated by $X$]
The \emph{subgroup generated by $X \subseteq G$} is the following subgroup, generated by taking the intersection of all subgroups containing $X$.
\[\Lambda = \{H \leq G : X \subseteq H\}\]
\[\langle X \rangle = \bigcap_{H \in \Lambda} H\]
\end{definition}


\begin{proposition}
\[ \langle X \rangle = \{ \prod_{i \in I} x_{i}^{n_i} : x_i \in X \land n_i \in \mathbb{N} \} \].
\end{proposition}


Inspired by this result, one may wish to find a way to generate a group with just some arbitrary set.

A \emph{word}
An \emph{empty word}
A \emph{reduced word}


An \emph{elementary word reduction algorithm}
Disjoint reductions $u_1 y_1 y^{-1}_{1} u_2  y_2 y^{-1}_{2} u_3$
start from either left or right
Overlapping reductions $u_1 y y^{-1} y u_2$
pair with left or right element

All Elementary word reduction algorithms have the same effect



A \emph{free group} $F(X)$ is a group generated by some set $X$ such that every non-empty reduced word is a nontrivial element of $F(X)$
\[\{x_i\}_{i=1}^{n} : [1,n]\cap \mathbb{N} \to X \cup X^{-1}\]
\[\{p_i\}_{i=1}^{n} : [1,n]\cap \mathbb{N} \to \mathbb{Z}\]
\[F(X) = \{ \prod_{i=1}^{n} x_{i}^{p_{i}} \}\]
This technique can be used for generating a group with one element too, however the definition given is usually more powerful.











Free groups
\section{Tietze transformations}
\begin{theorem}[Tietze's theorem]
\end{theorem}





\section{Lattices}

Lattices are a special subset of a linear space (usually $\mathbb{R}^n$) with these properties:
\begin{itemize}
\item Lattice points are closed under vector addition and subtraction
\item There is some minimum distance such that every lattice point is at least this distance from any other lattice point.
\item There is some maximum distance such that the distance between every point and their closest lattice point is less than this distance.
\end{itemize}


\[\Lambda = \{\sum^{n}_{i=1} a_i \mathbf{b}_i : a_i \in \mathbb{Z} \}\]

Lattices are isomorphic to free Abelian groups







\chapter{Torsion elements}

\begin{definition}[Torsion element]
\[g \text{ is a torsion element of } G \iff \exists n \in \mathbb{Z} [ g^n = 1_G ] \]
\end{definition}


We know that For finite groups, all elements are torsion elements. We've also been using torsion elements as a way to define cyclic subgroups. Where things get interesting is when we consider infinite groups where all elements are torsion elements.

\begin{definition}[Torsion group]
	\[G \text{ is a torsion group } \iff \forall g\in G [ g \text{ is a torsion element of } G ] \]
\end{definition}




\chapter{Abelienization}


When is a quotient group Abelian?

\begin{proposition}
$G / N$ is abelian iff $G' \subseteq N$
$G' = \{ [x,y] \in G : x,y \in G \}$
\begin{proposition}

This $G'$ is the \emph{commutator subgroup of $G$}.
