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:

1. $\textbf{[Closure]}$ for all $h,k\in H$ one has $h\cdot k\in H$, and
2. $(H,\cdot)$ is a group.

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 $\textdef{trivial subgroup}$.

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

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

<WRAP center round help 60%> 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 $(G,*)$ and $(G,m,e,i)$ definitions. </WRAP>

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

definition needsreview