\section{Proof of the Fundamental Theorem}

\begin{lemma}
	Let $G$ be a finite Abelian group of order $p^nm$, where $p$ is a prime that does not divide $m$. Then $G = H \times K$, where $H = \{x \in G\ \vert\ x^{p^n} =e\}$ and $K =\{x \in G\ \vert\ x^m = e\}$. Moreover, $\abs{H}=p^n$.
\end{lemma}

\begin{lemma}
	Let $G$ be an Abelian group of prime-power order and let $a$ be an element of maximum order in $G$. Then $G$ can be written in the form $\lr{a} \times K$.
\end{lemma}

\begin{lemma}
	A finite Abelian group of prime-power order is an internal direct product of cyclic groups.
\end{lemma}

\begin{lemma}
	Suppose that $G$ is a finite Abelian group of prime-power order. If $G=H_1 \times H_2 \times \dots \times H_m$ and $G=K_1 \times K_2 \times \dots \times K_n$, where the $H$'s and $K$'s are nontrivial cyclic subgroups with $\abs{H_1} \geq \abs{H_2} \geq \dots \geq \abs{H_m}$ and $\abs{K_1} \geq \abs{K_2} \geq \dots \geq \abs{K_n}$, then $m=n$ and $\abs{H_i} = \abs{K_i}$ for all $i$.
\end{lemma}
