Matrix groups and Symmetry
Linear algebra
See this video and these notes.
We use some facts from linear algebra:

A map $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is linear if $f(ax+by)=af(x)+bf(y)$ for all $x,y\in\mathbb{R}^{n}$ and all $a,b\in\mathbb{R}$.

An $n\times m$ matrix $A$ yields a linear map from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ via matrix multiplication $x\mapsto Ax$.

Given a linear map $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$, we can associate to it a matrix $A$ with entries $(a_{ij})$ by computing \(T(\mathbf{e}_j)= \sum_{i=1}^{m} a_{ij}\mathbf{f}_{i}\) where \(\mathbf{e}_{j}\) and \(\mathbf{f}_{i}\) are the $n$ and $m$ dimensional column vectors with a one in position $j$ (resp. $i$) and zeros elsewhere.

If $T:\mathbb{R}^{n}\to\mathbb{R}^{m}$ and $V:\mathbb{R}^{m}\to\mathbb{R}^p$ are linear maps with associated matrices $A$ and $B$, then the matrix associated to the composition $VT$ is the matrix product $BA$.

If $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is bijective then its inverse is also linear and the associated matrix is the inverse matrix $A^{1}$. Conversely if the associated matrix is invertible then $T$ is bijective.

An $n\times n$ matrix is invertible if and only if its determinant is nonzero.
Matrix Groups
See this video and these notes
The general and special linear groups
Definition: The general linear group of rank $n$, written $\mathrm{GL}_{n}(\mathbb{R})$, is the group of bijective linear transformations $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ with group operation given by composition of maps. Equivalently:
 $\mathrm{GL}_{n}(\mathbb{R})$ is the group of invertible $n\times n$ matrices with matrix multiplication
 $\mathrm{GL}_{n}(\mathbb{R})$ is the group of $n\times n$ matrices with nonzero determinant under matrix multiplication.
Definition: The special linear group \(\mathrm{SL}_{n}(\mathbb{R})\) is the subgroup of \(GL_{n}(\mathbb{R})\) of linear maps whose associated matrices have determinant one.
The orthogonal group
Definition: The Euclidean inner product on $\mathbb{R}^{n}$ is the dot product \((\sum_{i=1}^{n} a_{i}\mathbf{e}_{i})\cdot (\sum_{i=1}^{n} b_{i}\mathbf{e}_{i})=\sum_{i=1}^{n} a_{i}b_{i}.\) If $x,y\in\mathbb{R}^{n}$ this is also written $\langle x,y\rangle$.
The inner product is symmetric, bilinear, satisfies $x\cdot x\ge 0$ for all $x$, and $x\cdot x=0$ if and only if $x=0$.
Also $x\cdot x=x^2$ is the euclidean length of the vector $x$ and $(xy)\cdot (xy)=(xy)^2$ is the distance between $x$ and $y$.
Definition: A matrix $A\in\mathrm{GL}_{2}(\mathbb{R})$ is orthogonal if $Ax^2=x^2$ for all vectors $x$.
Proposition: $A$ is orthogonal if and only if $AA^{t}=Id$ or, equivalently, if the rows (or columns) of $A$ form an orthonormal set.
Definition: The orthogonal group $O_{n}(\mathbb{R})$ is the subgroup of \(\mathrm{GL}_{n}(\mathbb{R})\) consisting of orthogonal matrices. The special orthogonal group \(SO_{n}(\mathbb{R})\) is the subgroup of \(O_{n}(\mathbb{R})\) consisting of matrices with determinant $1$.
Extra topic (orientation)
See this video and these notes
The euclidean group
See this video and these notes
Definition: The euclidean group \(E_{n}(\mathbb{R})\) is the group of distance preserving maps \(f:\mathbb{R}^{n}\to\mathbb{R}^{n}\).
The Euclidean group can be described as the subgroup of $\mathrm{GL}_{n+1}(\mathbb{R})$ of matrices with the following block form:
\(\left(\begin{matrix} A & \Bigg\vert & \begin{matrix} x_1 \\ \vdots \\ x_n \end{matrix}\\ \hline \begin{matrix} 0 &\cdots & 0 \\ \end{matrix} &\vert &1\\ \end{matrix}\right)\)
where the $n\times n$ block $A$ is an orthogonal matrix.
If $M$ is an element of $E_n(\mathbb{R})$, and $y$ is a point in $\mathbb{R}^{n}$, then the transformation of $y$ by $M$ is given by matrix multiplication after extending $y$ by adding a $1$:
\[My = \begin{pmatrix} A & \Bigg\vert & \begin{matrix} x_1 \\ \vdots \\ x_n \end{matrix}\\ \hline \begin{matrix} 0 &\cdots & 0 \\ \end{matrix} &\Bigg\vert &1\\ \end{pmatrix}\begin{pmatrix} y_1 \\ \vdots \\ y_n \\ 1\end{pmatrix}\]Alternatively, $(A,x)y=Ay+x$.
Plane symmetries
See this video and these notes
By the definition $E(2)$ is the group of distance preserving maps of the plane to itself (the group of isometries).
Proposition: The group $SO(2)$ is abelian and consists of matrices of the form \(\left( \begin{matrix} a & b \\ b & a \\ \end{matrix}\right)\) where $a^2+b^2=1$ corresponding to rotations through an angle $\theta$ where \(\mathrm{cis}(\theta) = a+bi.\)
The group $O(2)$ includes $SO_{2}$ as well as matrices of the form \(\left( \begin{matrix} b & a \\ a & b \\ \end{matrix}\right)\) of determinant $1$.
Elements of $E(2)$ are rotations, reflections, translations, or “glide reflections.”
Definition: Two regions in the plane are congruent if there is an element of the Euclidean group $E(2)$ carrying one bijectively to the other.
Proposition: The only finite subgroups of $E(2)$ are isomorphic to \(\mathbb{Z}_{n}\) or \(\mathbb{D}_{n}\) for some $n\ge 1.$
Lattices
A lattice $L$ in the plane is the set of integer linear combinations of two independent vectors $v_1$, $v_2$: \(L =\{av_1+bv_2:a,b\in\mathbb{Z}\}.\)
Wallpaper and crystals
See these three videos:
and these notes
See also the interactive visualization at the bottom of this web page.
Definition: A wallpaper group $G$ is a subgroup of $E(2)$ that preserves a lattice; in other words, there is a lattice $L$ such that $gL=L$ for all $g\in G$.
Proposition: Up to isomorphism there are 17 wallpaper groups. Each such group yields a characteristic “wallpaper” made up of repeating patterns organized in a lattice shape.
This result is proved in the paper The 17 plane symmetry groups, by R. L. E. Schwarzenberger, The Mathematical Gazette, vol. 58, number 404, June 1974, pp. 123131. The techniques used are covered in this course, so the proof, though somewhat long, is accessible.
See this page for examples of the 17 groups and their associated patterns.
Definition: A crystallographic group $G$ is a subgroup of $E(3)$ that preserves a lattice. There are $219$ different crystallographic groups corresponding to $219$ different crystal structures. (Sometimes this number is given as $230$ depending on how you define “different.”)
Every possible crystal structure occurs for some mineral. See this page for a discussion.
You can see all possible plane symmetries here.