Do canonical transformations form a group?

Question

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical transformations comply with a group structure. But what should be the group operation? In other words, is there a notion of "product" under which it is closed and associative?

1. Existence of identity element: Identity transformation is canonical because the coordinates are mapped to themselves.
2. Existence of inverse: Inverse of a canonical transformation is canonical since the Poisson brackets are invariant.

Answer 1

There are, confusingly, different definitions of what exactly a "canonical transformation" is. See this answer by Qmechanic for an extended discussion and more useful links. The definition of a transformation in terms of canonical coordinates $q,p$ (a fixed "Darboux chart") does not suffice to talk about these transformations in full generality since the phase space of a generic system is not guaranteed to be covered by a single such chart.

If we either take the viewpoint that a canonical transformation is a symplectomorphism or an infinitesimal symplectomorphism as in this answer by Qmechanic, then the group operation is simply "composition" - carry out the two transformations in succession. Symplectomorphisms are a group under composition: Diffeomorphisms are a group under composition and it's easy to show that the condition for being a symplectomorphism is also fulfilled for the composition of two symplectomorphisms.