profinite completion

Last updated January 27, 2022

Idea

This construction takes any group and gives us an associated profinite group.

The part “completion” in the name perhaps refers to the fact that a group is dense in its profinite completion.

Definition

Let $G$ be an abstract group. Define the inverse system $(G/H, \phi_{HH'})$ as follows:

• $I={\text{finite index }H\triangleleft G}$ where $H\leq H'$ iff $H\supseteq H'$
• $\phi_{HH'}:G/H'\to G/H$ is the canonical projection map

The* profinite completion* is the inverse limit of this system: $$ \widehat{G}=\lim_\leftarrow G/H. $$

Properties

1. $\widehat{G}$ is a profinite group.
2. $G$ is dense in $\widehat{G}$.
3. The natural map $\gamma:G\to\widehat{G}$ is injective iff $G$ is residually finite (see profinite topology).

References

• https://www.quora.com/What-is-an-intuitive-explanation-of-profinite-completion-of-a-group