Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let $G_{\omega}$ denote the group of analytic functions from $M_{\omega}$ to itself which have an analytic inverse. Let $G_{\infty}, G_{r}$ be defined similarly.
My questions are
1.) What is known about the groups defined above ? (Anything about their structure, their interesting subgroups, or their homomorphic images etc.)
It appears to me that all the groups defined above are very big in general. For example in case $M=\mathbb{R}$, the group $G_{\omega}$ contains all maps of the type $x \to x+ \mu Sin (f(x))$ where $\mu$ is sufficiently small and $f(x)$ is analytic with bounded derivative. So it appears that this group is too big to be a finite dimensional Lie group.
2.) Is there a natural way to give these groups a manifold structure so as to make them 'infinite dimensional Lie groups' ?
3.) Are there conditions on $M$ which would ensure that any of these groups is a finite dimensional Lie group ?
If I am not making a mistake, in case $M=\mathbb{R}^n$, the action of each of these groups on $M$ is $p$-transitive where $p$ is any positive integer.
4.) Does it generalize to the general case where $M$ is not a Euclidean space ? Is the action of any of these groups 'infinity transitive' i.e. given two arbitrary sequences $\{ a_i\}$, $\{ b_i\}$ which have no limit point, does there exist an element of any of these groups which takes the sequence $\{ a_i\}$ to $\{ b_i\}$ ? Can we answer this question in specific cases (e.g. $M$ Euclidean) if not in general case ?
5.) One can look at the vector space of all real valued analytic/smooth/continuous functions defined on $M$. These vector spaces are also infinite dimensional in general. Is there a good description of the quotient of any of these vector spaces e.g. analytic functions modulo smooth functions ?
Please feel free to answer other natural related questions in these directions. It would be great if somebody could advise me some good references (books/articles/lecture notes) which addresses these or similar questions.
I had come across a book 'Continuous Groups of Transformations' by L.P.Eisenhart though am not sure what exactly it deals with and had asked a question about it here. Is this book a relevant reference for the kind of questions that I ask in this post ?
Some parts of this question are a bit vague, sorry about that.