Let be a group, and let be a nonempty subset of . Then is called a of if the following hold:

1. for all one has , and
2. is a group.

When is a subgroup of , we write . Any subgroup of that is not equal to itself is called a . The subset of consisting of just the identity is called the .

Let be a subset of a group . We say is a subgroup of if is a group itself when using the multiplication structure of restricted to . We'll write to mean is a subgroup of .

• This would be a good place to add links to any subgroup theorems.

Is this generic enough to work with both definitions of group that we know we'll be using? Do the words “restricted to” require a definition as well.

I think the definition I gave above could be improved. I tried to avoid talking about binary operations so we can use this for both the and definitions.

definition.subgroup.tex

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% DEPENDENCIES 
% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}} 
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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}

• Subgroup