Nilpotent action What is the definition of Nilpotent action of nilpotent group? Is the action of $Z_{2}$ on finitely generated abelian group is nilpotent action? 
 A: I think that a good reference here (at least in topological and homotopy theoretic setting) should be McCleary's book on spectral sequences.
There (chapter $\S8^{bis}.3$) you can find the following 
Definition:
Assume $G$-module $M$, then we can consider the "lower central series" of an action:
$$
  \ldots\subset\Gamma^r_GM\subset\Gamma^{r-1}_GM\subset\ldots\subset\Gamma^2_GM\subset M,
$$
where $\Gamma^2_G$ is a $\mathbb{Z}[G]$-submodule in M generated by elements of the form $gm-m,$ for $g \in G$ and $m \in M.$ Then $\Gamma^r_G$ is defined inductively as $\Gamma^2_G(\Gamma^{r-1}_GM).$ We then call an action of $G$ on $M$ nilpotent if this sequence becomes zero after some finite number of iterations.

In the same section of the book, you can find an example of a non-nilpotent action of $\mathbb{Z}/2$ on free abelian group.
To construct one you need to consider action on $\mathbb{Z}$ that changes sign.
$$
 1\mapsto -1, \text{ then }\quad \Gamma_{\mathbb{Z}/2}^2\mathbb{Z}=\mathbb{Z}\langle -1-1\rangle=2\mathbb{Z}
$$
It is then straightforward to show that $\Gamma_{\mathbb{Z}/2}^{r}=2^{r-1}\mathbb{Z}$ and thus an action is not nilpotent.
Topologically speaking this example corresponds to an action of the fundamental group $\pi_1(S^{2m}\times\mathbb{R}P^{2n})$ on n-th homotopy group $\pi_n(S^{2m}\times\mathbb{R}P^{2n})\cong\mathbb{Z}.$

Sidenote: I don't think that in order to have nilpotent action on needs to have a nilpotent group. Even if an action is effective (otherwise a counterexample is obvious). I don't know an immediate answer though.
