# On the Component Factor Group G/G_0 of a Pro-Lie Group G

## TL;DR

This paper proves that a pro-Lie group is almost connected if all its finite-dimensional Lie group quotients have finitely many components, filling a gap in the understanding of pro-Lie group structure.

## Contribution

It establishes a new characterization of almost connected pro-Lie groups based on their Lie group quotients, which was previously unproven.

## Key findings

- Pro-Lie groups are almost connected if all Lie group quotients have finitely many components.
- The proof involves verifying the completeness of the totally disconnected factor group.
- The result clarifies the relationship between pro-Lie groups and their finite-dimensional quotients.

## Abstract

A pro-Lie group $G$ is a topological group such that $G$ is isomorphic to the projective limit of all quotient groups $G/N$ (modulo closed normal subgroups $N$) such that $G/N$ is a finite dimensional real Lie group. A topological group is almost connected if the totally disconnected factor group $G_t:= G/G_0$ of $G$ modulo the identity component $G_0$ is compact. In this case it is straightforward that each Lie group quotient $G/N$ of $G$ has finitely many components. However, in spite of a comprehensive literature on pro-Lie groups, the following theorem, proved here, was not available until now: A pro-Lie group $G$ is almost connected if each of its Lie group quotients $G/N$ has finitely many connected components. The difficulty of the proof is the verification of the completeness of $G_t$.