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\begin{document}

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\section*{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]].

\hypertarget{derivations}{}\subsection*{{Derivations:}}\label{derivations}

Suppose $G$ is a [[group]] and $M$ a $G$-[[module]] and let $\delta : G \to M$ be a \textbf{derivation}. This means $\delta(g_1g_2) = \delta(g_1) +g_1\delta(g_2)$ for all $g_1, g_2 \in G$. (Note: \emph{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.

\begin{ulemma}
If $\delta : G \to M$ is a derivation, then

\begin{enumerate}%
\item $\delta(1_G) = 0$;


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


\item for any $g \in G$ and $n\geq 1$,

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


\end{enumerate}
\end{ulemma}
\begin{proof}
As was said, these are easy to prove.

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

\begin{displaymath}
\delta(1) = \delta(g^{-1}g) = \delta(g^{-1}) + g^{-1}\delta(g)
\end{displaymath}
to get (2), and finally induction to get (3).

\end{proof}
\hypertarget{remarks_and_examples}{}\subsection*{{Remarks and examples:}}\label{remarks_and_examples}

\begin{itemize}%
\item There is a mapping from $G$ to its [[augmentation ideal]], $I(G)$, defined by $d_G(g)= g-e_G$. This is the [[derived module|universal derivation]] towards $G$-modules.


\item 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).



\end{itemize}
\hypertarget{example_calculation_using_the_fox_derivative_wrt_a_generator}{}\subsubsection*{{Example calculation using the [[Fox derivative]] w.r.t a generator.}}\label{example_calculation_using_the_fox_derivative_wrt_a_generator}

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

\begin{displaymath}
\frac{\partial r}{\partial u} = 1 + u v - u v u v^{-1} u^{-1},
\end{displaymath}
\begin{displaymath}
\frac{\partial r}{\partial v} = u - u v u v^{-1} -  u v u v^{-1} u^{-1} v^{-1}.
\end{displaymath}
This relation, $r$, is the typical [[braid group]] relation, here in $Br_3$.

\hypertarget{relative_derivations}{}\subsection*{{Relative derivations}}\label{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 $\varphi$.

Let $\varphi  : H \rightarrow  G$ be a homomorphism of groups. A \emph{$\varphi$-derivation} from a group to a module,

\begin{displaymath}
\partial  : H \rightarrow  M,
\end{displaymath}
from $H$ to a left $\mathbb{Z}[G]$-module, $M$, is a mapping from $H$ to $M$, which satisfies the equation

\begin{displaymath}
\partial (h_1 h_2 ) = \partial (h_1 ) + \varphi (h_1)\partial (h_2 )
\end{displaymath}
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 [[Masanori Morishita|Morishita]].

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

\hypertarget{references}{}\subsection*{{References}}\label{references}

For the original version of derived module, see

\begin{itemize}%
\item [[R. H. Crowell]], \emph{The derived module of a homomorphism}, Advances in Math., 5, (1971), 210--238.

\end{itemize}
For applications in [[arithmetic topology]]

\begin{itemize}%
\item [[Masanori Morishita]], \emph{Knots and Primes: An Introduction to Arithmetic Topology}, 2012 (\href{https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false}{web})

\end{itemize}


\end{document}