derivation on a group



For convenience we assume below that MM is a GG-module, it does not in general have to be abelian and it suffices to have it a GG-group.

Suppose GG is a group and MM a GG-module and let δ:GM\delta : G \to M be a derivation in the sense that δ(g 1g 2)=δ(g 1)+g 1δ(g 2)\delta(g_1g_2) = \delta(g_1) +g_1\delta(g_2) for all g 1,g 2Gg_1, g_2 \in G (a crossed homomorphism – notice that it’s not δ(g 1)g 2+g 1δ(g 2)\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 δ:GM\delta : G \to M is a derivation, then

  1. δ(1 G)=0 \delta(1_G) = 0;

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

  3. for any gGg \in G and n1n\geq 1,

    δ(g n)=( k=0 n1g k)δ(g).\delta(g^n) = (\sum^{n-1}_{k=0}g^k)\delta(g).

As was said, these are easy to prove.

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

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

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

Remarks and examples:

Example calculation using the Fox derivative w.r.t a generator.

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

ru=1+uvuvuv 1u 1, \frac{\partial r}{\partial u} = 1 + u v - u v u v^{-1} u^{-1},
rv=uuvuv 1uvuv 1u 1v 1. \frac{\partial r}{\partial v} = u - u v u v^{-1} - u v u v^{-1} u^{-1} v^{-1}.

This relation, rr, is the typical braid group relation, here in Br 3Br_3.

Relative derivations

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 φ\varphi.

Let φ:HG\varphi : H \rightarrow G be a homomorphism of groups. A φ\varphi -derivation from a group to a module,

:HM,\partial : H \rightarrow M,

from HH to a left [G]\mathbb{Z}[G]-module, MM, is a mapping from HH to MM, which satisfies the equation

(h 1h 2)=(h 1)+φ(h 1)(h 2)\partial (h_1 h_2 ) = \partial (h_1 ) + \varphi (h_1)\partial (h_2 )

for all h 1h_1, h 2Hh_2 \in H.

There is a universal such φ\varphi-derivation, d φ:HD φd_\varphi:H\to D_\varphi. The codomain of this is variously called the derived module of φ\varphi (e.g. by Crowell) or the φ\varphi-differential module by Morishita.

The set of φ\varphi-derivations is often written Der φ(H,M)Der_\varphi(H,M), or simply Der φ(M)Der_\varphi(M).


For the original version of derived module, see

Textbook account:

Application in arithmetic topology: