On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$

TL;DR

This paper constructs minimal solvable and verbal embeddings of the rational numbers' additive group into 2-generator groups, preserving order and subnormality, advancing understanding of group embeddings.

Contribution

It demonstrates the existence of order-preserving embeddings of  into minimal solvable and verbal 2-generator groups, with specific structural properties.

Findings

Embedding of  into a solvable group of length 3

Existence of order-preserving verbal embeddings for any non-trivial word set

Embeddings are subnormal

Abstract

We show that there is an order-preserving embedding of the additive group of rational numbers $\mathbb{Q}$ into a 2-generator group $G$. The group $G$ can be chosen to be a solvable group $G$ of length 3, which is a minimal result in the sense that it cannot be chosen to be neither solvable of length 2, nor a nilpotent group. For any non-trivial word set $V \subseteq F_\infty$ there is an order-preserving verbal embedding of $\mathbb{Q}$ into a 2-generator group $G$. The embeddings constructed are subnormal.