TITLE: Nice examples of groups which are not obviously groups
QUESTION [319 upvotes]: 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.

REPLY [2 votes]: The set of reversible Turing machines forms a group: https://arxiv.org/pdf/1603.08715.pdf
But, since they define Turing machines not in the classical way, but in a way that composition is an operation and then take the reversible ones, it doesn't feel like it is so difficult to see that it is a group. 
I find it very interesting, thou!