TITLE: Theorems for nothing (and the proofs for free)
QUESTION [50 upvotes]: Some theorems give far more than you feel they ought to: a weak hypothesis is enough to prove a strong result.  Of course, there's almost always a lot of machinery hidden below the waterline.  Such theorems can be excellent starting-points for someone to get to grips with a new(ish) subject: when the surprising result is no longer surprising then you can feel that you've gotten it.
Let's have some examples.

REPLY [9 votes]: The only group with order $p$ a prime is $\mathbb{Z}/p\mathbb{Z}$