# nLab nilpotent module

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

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

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

$\Gamma_0 N \supset \Gamma_1 N \supset \Gamma_2 N \supset \cdots \,,$

given recursively by

\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 $k_{max} \in \mathbb{N}$ with $\Gamma_{k_{max}}N = 0$.

## References

• Peter Hilton, Nilpotency in group theory and topology, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 (jstor:43741908)