For convenience we assume below that $M$ is a $G$-module, it does not in general have to be abelian and it suffices to have it a $G$-group.
Suppose $G$ is a group and $M$ a $G$-module and let $\delta : G \to M$ be a derivation in the sense that $\delta(g_1g_2) = \delta(g_1) +g_1\delta(g_2)$ for all $g_1, g_2 \in G$ (a crossed homomorphism – notice that it’s not $\delta(g_1)g_2 + g_1\delta(g_2)$ as for the other notion of derivation.)
For calculations, the following lemma is very valuable, although very simple to prove.
If $\delta : G \to M$ is a derivation, then
$\delta(1_G) = 0$;
$\delta(g^{-1}) = -g^{-1}\delta(g)$ for all $g \in G$;
for any $g \in G$ and $n\geq 1$,
As was said, these are easy to prove.
$\delta(g) = \delta(1) + 1\delta(g)$, so $\delta(1)= 0$, and hence (1); then
to get (2), and finally induction to get (3).
There is a mapping from $G$ to its augmentation ideal, $I(G)$, defined by $d_G(g)= g-e_G$. This is the universal derivation towards $G$-modules.
The Fox derivatives are examples of derivations. It is worth noting that this lemma allows a simplification of the conditions given there (as noted there).
Let $X = \{u,v\}$, with $r \equiv u v u v^{-1} u^{-1} v^{-1} \in F = F(u,v),$ then
This relation, $r$, is the typical braid group relation, here in $Br_3$.
These are a useful relative form of derivation. The notion is often avoided as it can easily be reduced to the more standard form above by restricting the module structure along $\varphi$.
Let $\varphi : H \rightarrow G$ be a homomorphism of groups. A $\varphi$-derivation from a group to a module,
from $H$ to a left $\mathbb{Z}[G]$-module, $M$, is a mapping from $H$ to $M$, which satisfies the equation
for all $h_1$, $h_2 \in H$.
There is a universal such $\varphi$-derivation, $d_\varphi:H\to D_\varphi$. The codomain of this is variously called the derived module of $\varphi$ (e.g. by Crowell) or the $\varphi$-differential module by Morishita.
The set of $\varphi$-derivations is often written $Der_\varphi(H,M)$, or simply $Der_\varphi(M)$.
For the original version of derived module, see
Textbook account:
Application in arithmetic topology: