TITLE: What's an easy way of proving a subgroup is normal?
QUESTION [19 upvotes]: I think in most situations(for example, in $S_n$ or $D_n$), proving by definition is too complicated because you have to calculate $gng^{-1}$ for every $n$ in $N$ and $g$ in $G$. To prove that all the left cosets are also right cosets is also too complicated because you have to find all those cosets. I wonder if there's a way to do this without having to calculate everything by hand.

REPLY [2 votes]: If you know a subgroup of a particular order is the ONLY subgroup of that order, then you know it's normal. I know that's a unique case, but just another tool to keep in mind.