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: \DeclareMathOperator{\syl}{Syl} 
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\begin{definition}
Please add comments above to adjust this definition until we come upon one we agree on.
\end{definition}

• Subgroup

definition