Are there nonlinear operators that have the group property? To be clear: What I am actually talking about is a nonlinear operator on a finitely generated vector space V  with dimension $d(V)\;\in \mathbb{N}>1$. I can think of several nonlinear operators on such a vector space but none of them have the requisite properties of a group. In particular, but not exclusively, are there any such nonlinear operator groups that meet the definition of a Lie Group. 

How about the following construction. Let $M \in R_{d}(G)$ be a $d$ dimensional representation of some Lie group (e.g. $SL(2,R)$ and $V$ is some $d$ dimensional vector space. Now define:
$$
f(x,M,\epsilon)= \frac{M x}{(1+\epsilon w.x)}
$$
where the vector $w$ is chosen so that $w.M= w$ for all $M\in SL(2,\mathbb{R})$. Such vectors exist, for instance, if $d=4$ and $R_4(G)=R_2(G) \otimes R_2(G)$ then $w=\begin{pmatrix}
0 \\
1\\
-1\\
0
\end{pmatrix}$ is one such vector. It is now not too difficult to convince oneself that:
$$ \begin{eqnarray}
f(x,I,0) = x \\
f(f(x,M_1,\epsilon_1),M_2,\epsilon_2) = f(x,M_1 M_2,\epsilon_1+\epsilon_2)
\end{eqnarray}$$
so that the mapping $f$ is nonlinear and satisfies the group property.
