On p-adic Integers and The Adding Machine Group

TL;DR

This paper demonstrates that the p-adic integers can be isometrically embedded into the automorphism group of a rooted tree, specifically through the closure of the adding machine group, revealing a deep connection between p-adic number theory and automorphism groups.

Contribution

It establishes an isometric and isomorphic embedding of the p-adic integers into the automorphism group of a rooted tree via the closure of the adding machine group.

Findings

The closure of the adding machine group is isometric to the p-adic integers.

The p-adic integers can be embedded into the automorphism group of a rooted tree.

The group of p-adic integers is isomorphic to a subgroup of Aut(X*).

Abstract

In this paper, we define a natural metric on Aut(X*) and prove that the closure of the adding machine group, a subgroup of the automorphism group, is both isometric and isomorphic to the group of p-adic integers. So, we show that the group of p-adic integers can be isometrically embedded into the metric space Aut(X*).