derivation on a group
[For convenience we assume below that is a -module, it does not in general have to be abelian and it suffices to have it a -group.]
Suppose is a group and a -module and let be a derivation. This means for all . (Note: not as for the other notion of derivation.)
For calculations, the following lemma is very valuable, although very simple to prove.
If is a derivation, then
for all ;
for any and ,
As was said, these are easy to prove.
, so , and hence (1); then
to get (2), and finally induction to get (3).
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 , with then
This relation, , is the typical braid group relation, here in .
Revised on October 9, 2012 13:36:47
by Tim Porter