# Centric linking systems of locally finite groups

## TL;DR

This paper introduces the concept of centric linking systems for locally finite groups and proves a homotopy equivalence between the p-completion of the classifying space and the nerve of the linking system under certain conditions.

## Contribution

It defines centric linking systems for locally finite groups and establishes a homotopy equivalence result for their classifying spaces under specific countability conditions.

## Key findings

- Defined centric linking systems for locally finite groups.
- Proved homotopy equivalence between classifying space p-completion and nerve of linking system.
- Established conditions under which the equivalence holds.

## Abstract

These notes are defining the notion of centric linking system for a locally finite group If a locally finite group $G$ has countable Sylow $p$-subgroups, we prove that, with a countable condition on the set of intersections, the $p$-completion of its classifying space is homotopy equivalent to the $p$-completion of the nerve of its centric linking system.