\section{Lagrange's Theorem and Consequences}

\begin{theorem}[Lagrange's Theorem: $\mathbf{\abs{H} \text{ Divides } \abs{G}}$]
	If $G$ is a finite group and $H$ is a subgroup of $G$, then $\abs{H}$ divides $\abs{G}$. Moreover, the number of distinct left (right) cosets of $H$ in $G$ is $\abs{G}/\abs{H}$.
\end{theorem}

\begin{remark}
	A special name and notation have been adopted for the number of left (or right) cosets of a subgroup in a group. The \textit{index} of a subgroup $H$ in $G$ is the number of distinct left cosets of $H$ in $G$. This number is denoted by $\abs{G:H}$.
\end{remark}

\begin{corollary}[$\mathbf{\abs{G:H} = \abs{G}/\abs{H}}$]
	If $G$ is a finite group and $H$ is a subgroup of $G$, then $\abs{G:H} = \abs{G}/\abs{H}$.
\end{corollary}

\begin{corollary}[$\mathbf{\abs{a}}$ Divides $\mathbf{\abs{G}}$]
	In a finite group, the order of each element of the group divides the order of the group.
\end{corollary}

\begin{corollary}[Groups of Prime Order Are Cyclic]
	A group of prime order is cyclic.
\end{corollary}

\begin{corollary}[$\mathbf{a^{\abs{G}}=e}$]
	Let $G$ be a finite group, and let $a \in G$. Then, $a^{\abs{G}} = e$.
\end{corollary}

\begin{corollary}[Fermat's Little Theorem]
	For every integer $a$ and every prime $p$, $a^p \mod p = a \mod p$.
\end{corollary}

\begin{theorem}[$\mathbf{\abs{HK} = \abs{H}\abs{K}/\abs{H \cap K}}$]
	For two finite subgroups $H$ and $K$ of a group, define the set $HK = \{hk\ \vert\ h \in H, k \in K\}$. Then $\abs{HK} = \abs{H}\abs{K}/\abs{H \cap K}$.
\end{theorem}

\begin{theorem}[Classification of Groups of order 2$\mathbf{p}$]
	Let $G$ be a group of order $2p$, where $p$ is a prime greater than 2. Then $G$ is isomorphic to $\Z_{2p}$ or $D_p$.
\end{theorem}
