Nice examples of groups which are not obviously groups 
I am searching for some groups, where it is not so obvious that they are groups.

In the lecture's script there are only examples like $\mathbb{Z}$ under addition and other things like that. I don't think that these examples are helpful to understand the real properties of a group, when only looking to such trivial examples. I am searching for some more exotic examples, like the power set of a set together with the symmetric difference, or an elliptic curve with its group law. 

I always found the fact that braid groups are groups at all quite interesting. The elements of the group are all the different braids you can make with, say, $n$ strings.  The group operation is concatenation.  The identity is the untangled braid.  But the fact that inverses exist is not obvious.
