nLab
derivation on a group

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. This means $\delta (g_1{g}_2)=\delta (g_1)+g_1\delta (g_2)$ for all $g_1,g_2\in G$. (Note: 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.

Lemma

If $\delta :G\to M$ is a derivation, then

1. $\delta (1_G)=0$;

2. $\delta (g^{-1})=-g^{-1}\delta (g)$ for all $g\in G$;

3. for any $g\in G$ and $n\ge 1$,

   $$\delta (g^n)=(\sum _{k=0}^{n-1}{g}^k)\delta (g).$$\delta(g^n) = (\sum^{n-1}_{k=0}g^k)\delta(g).

Proof

As was said, these are easy to prove.

$\delta (g)=\delta (1g)+1\delta (g)$, so $\delta (1)=0$, and hence (1); then

$$\delta (1)=\delta (g^{-1}g)=\delta (g^{-1})+g^{-1}\delta (g)$$\delta(1) = \delta(g^{-1}g) = \delta(g^{-1}) + g^{-1}\delta(g)

to get (2), and finally induction to get (3).

Remark:

The Fox derivatives are examples. It is worth noting that this lemma allows a simplification of the conditions given there (as noted there).

Example (using the Fox derivative w.r.t a generator)

Let $X=\{u,v\}$, with $r\equiv uvu{v}^{-1}{u}^{-1}{v}^{-1}\in F=F(u,v),$ then

$$\frac{\partial r}{\partial u}=1+uv-uvu{v}^{-1}{u}^{-1},$$\frac{\partial r}{\partial u} = 1 + u v - u v u v^{-1} u^{-1},

$$\frac{\partial r}{\partial v}=u-uvu{v}^{-1}-uvu{v}^{-1}{u}^{-1}{v}^{-1}.$$\frac{\partial r}{\partial v} = u - u v u v^{-1} - u v u v^{-1} u^{-1} v^{-1}.

This relation, $r$, is the typical braid group relation, here in ${\mathrm{Br}}_3$.