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 was surprised to learn about the elliptic curve groups.  You fix constants $a$ and $b$ take the set $S$ of points on the Riemann sphere (that is, the complex plane plus a point at infinity) of the form $$y^2 = x^3 + ax + b.$$ Then define the sum two points $p_1, p_2$ on this curve by taking the straight line through $p_1$ and $p_2$ and finding the third point $p_3 = \langle x_3, y_3\rangle$ where the line intersects $S$. Then $p_3^{-1} = \langle x_3, -y_3\rangle$ is the group sum of $p_1$ and $p_2$.  It's not immediately clear that there is necessarily a point $p_3$, but there is, with suitable treatment of tangents and of the point at infinity. It's not immediately clear that the operation is associative, but it is. The point at infinity is the identity element, and the inverse of the point $\langle x, y\rangle$ is $\langle x, -y\rangle $.


