Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-4/classification-of-subgroups-of-cyclic-groups.tex

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+\section{Classification of Subgroups of Cyclic Groups}
+
+\begin{theorem}[Fundamental Theorem of Cyclic Groups]
+	Every subgroup of a cyclic group is cyclic. Moreover, if $\abs{\lr{a}} = n$, then the order of any subgroup of $\lr{a}$ is a divisor of $n$; and, for each, positive divisor $k$ of $n$, the group $\lr{a}$ has exactly one subgroup of order $k$ -- namely, $\lr{a^{n/k}}$.
+\end{theorem}
+
+\begin{corollary}[Subgroups of $\mathbf{\Z_n}$]
+	For each positive divisor $k$ of $n$, the set $\lr{n/k}$ is the unique subgroup of $\Z_n$ of order $k$; moreover, these are the only subgroups of $\Z_n$.
+\end{corollary}
+
+\begin{theorem}[Number of Elements of Each Order in a Cyclic Group]
+	If $d$ is a positive divisor of $n$, the number of elements of order $d$ in a cyclic group of order $n$ is $\phi(d)$.
+\end{theorem}
+
+\begin{corollary}[Number of Elements of Order $\mathbf{d}$ in a Finite Group]
+	In a finite group, the number of elements of order $d$ is a multiple of $\phi(d)$.
+\end{corollary}