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. 

Sandpile groups have the curious feature that the identity element is very complicated. See "What is a Sandpile?" (Levine, Lionel and Propp, James.  Notices of the AMS, 57 #8 (September 2010) pp. 976–979).

