If we start with general (invertable) functions $f:\mathbb{R}^N \rightarrow \mathbb{R}^N$. What conditions can we put on them such that they form a group. I can think of a few:
We could restrict the space to a surface $S \subset \mathbb{R}^N$ so that $f:S\rightarrow S$. For example f could be diffeomorphisms of the sphere. So for $f(x)$ we would have $|f|^2=1$ and $|x|^2=1$. We could restrict S to be only affine transformations then we would get the Lie groups such as O(N). Or with translations it would be the Poincaré group. $f(x)=P+Mx$, with $M^TM=1$.
We could pair up dimensions and functions $\{f(x,y),g(x,y)\}$ and treat them as holomorphic functions. $z=x+iy, h=f+ig$. With $\partial_x f - \partial_y g=0$ and $\partial_y f + \partial_x g=0$ etc. So this reduces the function space from $\mathbb{R}^{2N}\rightarrow \mathbb{R}^{2N}$ to $\mathbb{Z}^N \rightarrow \mathbb{Z}^N$. (What about for quaternions?)
We could restrict $f$ to be conformal angle preserving functions. $f(x)=(ax+b)/(cx+d)$ and its multidimensional generalisations. This is equivalent to the condition $2 f'''(x) f'(x) - 3f''(x)^2=0$
I can't think of any more. It seems to me that most can be derived by specifying some differential equations $L[f]=0$ so that it holds $L[f(g)]=0$ also. Or by specifying a restriction on the space as a subsurface such has $s(x)=0$.
Is there a general classification? (What about also for superspace?)