# nLab completion of a group

Contents

Not to be confused with group completion.

group theory

# Contents

(…)

## Definition

###### Definition

($R$-complete group)
For $R$ a solid ring, a group $G$ is called $R$-complete if it has a central series

$G \,=\, G_0 \supset G_1 \supset \cdots \supset G_k \,=\, 1$

(i.e. such that under commutator subgroups $[-,-]$ we have $[G, G_i] \subset G_{i+1}$)

such that each quotient group $G_j/G_{j+1}$ (which is an abelian group by the previous property) carries the structure of an $R$-module.

###### Proposition

(completion of a group)
Given a solid ring $R$ and any group $G$, there is a universal construction of an $R$-complete group $G \xrightarrow{\;\;} G_{\widehat{R}}$ – the $R$-completion of $G$ – given by the limit

$G_{\widehat{R}} \;\simeq\; \underset{ \underset{ { G \to C \in } \atop { Grp^{G/}_{/R CompGrp} } }{\longleftarrow} }{\lim} \;C\;$

over the codomain functor from the comma category of $R$-complete groups under $G$.

More generally:

###### Definition

(fiberwise $R$-complete group)
A short exact sequence of groups

(1)$1 \to N \longrightarrow \widehat{G} \longrightarrow G \to 1$

is fiberwise or relative $R$-complete if $N$ is so (in the sense of Def. ).

###### Proposition

(fiberwise completion of a group)
Given a solid ring $R$ and any short exact sequence of groups (1), there is a universal construction of a fiberwise $R$-complete exact sequence

$\array{ 1 &\to& N &\longrightarrow& \widehat{G} &\longrightarrow& G &\to& 1 \\ && \big\downarrow && \big\downarrow && \big\downarrow \mathrlap{{}^{=}} \\ 1 &\to& N_{\widehat{R}} &\longrightarrow& \widehat{G}_{\widehat{R}_G} &\longrightarrow& G &\to& 1 }$

## Properties

### Special cases

###### Example

(profinite completion)
For $R = \mathbb{Z}/p$ (a cyclic group of prime order) and applied to finitely generated groups, $R$-completion coincides with $p$-profinite completion.

###### Example

(Malcev completion)
For $R = \mathbb{Q}$ (the rational numbers) and applied to nilpotent groups, $R$-completion coincides with Malcev completion.

## References

Original articles:

Further developments:

Last revised on July 16, 2022 at 21:21:09. See the history of this page for a list of all contributions to it.