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\begin{definition}
Let $(G,\cdot)$ be a group, and let $H$ be a nonempty subset of $G$. Then $H$ is called a \textdef{subgroup} of $G$ if the following hold:
\begin{enumerate}
\item \textbf{[Closure]} for all $h,k\in H$ one has $h\cdot k\in H$, and
\item  $(H,\cdot)$ is a group.
\end{enumerate}
When $H$ is a subgroup of $G$, we write $H\le G$.  Any subgroup of $G$ that is not equal to $G$ itself is called a \textdef{proper subgroup}. The subset of $G$ consisting of just the identity is called the \textdef{trivial subgroup}.
\end{definition}