nilpotent module



Let GG be a group, NN an abelian group and ()():G×NN(-)\cdot (-) \colon G \times N \to N an action of GG on NN by linear maps, thus making NN a module over GG.

Then this is called a nilpotent module if the sequence of abelian subgroups

Γ 0NΓ 1NΓ 2N, \Gamma_0 N \supset \Gamma_1 N \supset \Gamma_2 N \supset \cdots \,,

given recursively by

Γ 0N N Γ k+1N {gnn|gG,nΓ nN}, \begin{aligned} \Gamma_0 N & \coloneqq\; N \\ \Gamma_{k+1} N & \coloneqq\; \big\{ g \cdot n - n \;\vert\; g \in G,\; n \in \Gamma_n N \big\} \,, \end{aligned}

terminates, in that there is k maxk_{max} \in \mathbb{N} with Γ k maxN=0\Gamma_{k_{max}}N = 0.