Are there any interesting examples of functions on Abelian groups that are not homomorphisms?

Question

It seems that most maps on (abelian) groups $G$ studied are homomorphisms, that is they obey $\phi(ab)=\phi(a)\phi(b)$ for $a,b\in G$.

Are there any interesting examples of maps $\phi:G\to H$ in the literature that are not homomorphisms (but still have some structure)?

(I am mostly interested in abelian groups. Hence that rules out the "antihomomorphism" $\phi(ab)=\phi(b)\phi(a)$ in the case of abelian groups.)

Thanks.

Answer 1

There are the quasi-homomorphisms, which are maps $f\colon G\longrightarrow H$ such that the set$$\{g,g'\in G\,|\,f(g+g')-f(g)-f(g')\}$$is finite. Of course, if $G$ is finite, that any map from $G$ into $H$ is a quasi-homomorphism. On the other hand, the quasi-homomorphisms from $(\mathbb{Z},+)$ into itself can be used to construct the real numbers.

Answer 2

Let $G$ be a group (not necessarily abelian), and $V$ a vector space with a linear action of $G$ (thus, a representation of $G$). Consider derivations $d : G \rightarrow \operatorname{End}(V)$ (the latter is the algebra of linear operators), satisfying the following relations:

$$d(1_G)=\operatorname{Id}$$ $$d(g\cdot h)=d(g)\cdot h + g \cdot d(h)$$

(the second relation is usually called the Leibniz rule).

Usually, though, this is studied in a more general context of a ring $R$ and an $R$-module $M$, with derivation $d:R \rightarrow \operatorname{End}(M)$.

Answer 3

$GL_n(\mathbb R)$ is a group under matrix multiplication and define a map $\operatorname{tr}:GL_n(\mathbb R) \to \mathbb R$ by $A\to \operatorname{tr}(A)$. It is not a group homomorphism because it is not multiplicative: $\operatorname{tr}(AB)\ne \operatorname{tr}(A)\operatorname{tr}(B)$.