Hilbert's Fifth Problem for Local Groups

TL;DR

This paper proves that every locally Euclidean local group is locally isomorphic to a Lie group, resolving Hilbert's fifth problem for local groups using nonstandard analysis techniques.

Contribution

It provides a rigorous proof of Hilbert's fifth problem for local groups, correcting previous flawed claims and applying nonstandard analysis methods.

Findings

Every locally Euclidean local group is locally isomorphic to a Lie group

The proof corrects and completes the previous flawed proof by Jacoby

Uses nonstandard analysis to establish the result

Abstract

We solve Hilbert's fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert's fifth problem for global groups by Hirschfeld.