# Do Legendre transformation form a group?

In my classical mechanics class, my professor asked if Legendre transformations form a group, and in my little knowledge about groups, I know that a transformation group consists of a set of transformations and compositions of transformation as the group operation. The first "problem" I encounter was that, the way I see it and I don't know if it is correct, is that there's only "one" legendre transformation, so my first thought was that this set would consist only of one element, and knowning that if I apply two times the legendre transformation to a function, it gives me the same function, It would be its own inverse element. But then I have problems thinking about the identity transformation, so maybe it could form a group if we consider the set of the legendre transformation plus the identity transformation. As i said before, I'm not an expert in group theory, I just started learning about groups, so if i said something incorrect, I would be grateful if you correct me. Thank you.

Think about a system with the so-called $$C_2$$ rotational axis. That is a system such that a rotation by an angle $$\pi$$ around that axis corresponds to a configuration indistinguishable from the original.
The set of rotations around such $$C_2$$ axis has the algebraic structure of a group of two elements, The identity $$E$$, i.e., all the equivalent rotations by $$2 \pi n$$, with $$n$$ integer, and a $$R_2$$ rotation, i.e., all the equivalent rotations by $$\pi n$$, with $$n$$ odd integer.