Let V be a finitely generated vector space with dimension(V) = $n \in \mathbb{N}>1$. Now let T: $ V \to V$ be a map such that $\forall \hat{v},\hat{w} \in V$, $\; T(\hat{v}+\hat{w}) \neq T(\hat{v})+T(\hat{w})$.So if $\chi(V,V)$ be the collection of mappings with the property that T has, then what I'm looking for is a subset $G \subset \chi(V,V) $ such that there is a function $\lambda: G \times G \to G$ and a function ${i}: G \to G $ with the following (4) properties:


1. $\forall A,B \in G,\; \lambda(A,B) \in G \;$ (*closure*)
2. $\exists E \in G$ such that $\forall A \in G,\; \lambda(A,E)=\lambda(E,A)=A \;$ (*identity*)
3. $\forall A \in G,\; \exists A^{-1} \in G \;$ such that $\lambda(A, A^{-1})=\lambda( A^{-1},A)=E \; $ (*inverse*)
4. $\forall A,B,C \in G, \; \lambda(A,\lambda(B,C))= \lambda(\lambda(A,B),C) \;$ (*associativity*)


And to clarify, ${i}(A) = A^{-1} \; (\forall A \in G)$. Can someone show an example(or two)of such a collection of operators(and a binary function) that satisfy these properties?



Nonlinear Operators(with the group property?)
Fix $a\in V, a\ne 0$ and for $L\in {\rm Hom}(V,V)$ denote $L\_a:x\to L(x)+a$. Let $G$ to be the set of all $L\_a (L\in {\rm Hom}(V,V)$ and $$\lambda (L\_a,M\_a)(x) =L\_a(x)+M\_a(x)-a=(L+M)(x)+a.$$


