The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's [*A History of Mathematics*](http://books.google.com/books?id=xwIZQwAACAAJ&dq=history+of+mathematics&hl=en&ei=p-h_TPeRDYWclgeYu8zBDg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CFkQ6AEwBg) off the bookshelf and was surprised to find that *isomorphism* isn't even listed in its index. The [Wikipedia article on isomorphisms](http://en.wikipedia.org/wiki/Isomorphism) only gives two concrete examples.

There are many surprising, significant, classic isomorphisms. I'll refrain from giving examples. What are your favorites?

As usual, please limit yourself to **one isomorphism per answer**.

*(Related: [your favorite surprising connections in mathematics](http://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics). But this question is looking for more concrete examples, particularly those that illustrate the power of the idea.)*