Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-28/chapter-28.tex

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+\chapter{Frieze Groups and Crystallographic Groups}
+\subimport{./}{the-frieze-groups.tex}
+\subimport{./}{the-crystallographic-groups.tex}
+\subimport{./}{identification-of-plane-periodic-patterns.tex}

part-5/chapters/chapter-28/identification-of-plane-periodic-patterns.tex

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+\section{Identification of Plane Periodic Patterns}
+
+\begin{remark}
+	A \textit{lattice of points} of a pattern is a set of images of any particular point acted on by the translation group of the pattern. A \textit{lattice unit} of a pattern whose translation subgroup is generated by $u$ and $v$ is a parallelogram formed by a point of the pattern and its image under $u,v$, and $u + v$. A \textit{generating region} (or \textit{fundamental region}) of a periodic pattern is the smallest portion of the lattice unit whose images under the full symmetry of the group of the pattern cover the plane.
+\end{remark}

part-5/chapters/chapter-28/the-crystallographic-groups.tex

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+\section{The Crystallographic Groups}
+
+\begin{remark}
+	The seven frieze groups catalog all symmetry groups that leave a design invariant under all multiples of just one translation. However, there are 17 additional kinds of discrete plane symmetry groups that arise from infinitely repeating designs in a plane. these groups are the symmetry groups of plane patterns whose subgroups of translations are isomorphic to $\Z \oplus \Z$. Consequently, the patterns are invariant under linear combinations of two linearly independent translations. These 16 groups were first studied by the 19th-century crystallographers and often called the \textit{plane crystallographic groups}. Another term occasionally used for these groups is \textit{wallpaper groups}.
+\end{remark}

part-5/chapters/chapter-28/the-frieze-groups.tex

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+\section{The Frieze Groups}
+
+\begin{remark}
+	In this chapter, we discuss an interesting collection of infinite symmetry groups that arise from periodic designs in a plane. There are two types of such groups. The \textit{discrete frieze groups} are the plane symmetry groups of patterns whose subgroups of translations are isomorphic to $\Z$. These kinds of designs are the ones used for decorative strips and for patterns on jewelry. In mathematics, familiar examples include the graphs of $y=\sin(x)$, $y=\tan(x)$, $y=\abs{\sin(x)}$, and $\abs{y} = \sin(x)$. After we analyze the discrete frieze groups, we examine the discrete symmetry groups of plane patterns whose subgroups of translations are isomorphic to $\Z \oplus \Z$.
+\end{remark}