On Order-Preserving and Verbal Embeddings of the Group $\mathbb{Q}$
Arman Darbinyan, Vahagn H. Mikaelian

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 into a 2-generator group . The group can be chosen to be a solvable group 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 there is an order-preserving verbal embedding of into a 2-generator group . The embeddings constructed are subnormal.
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Taxonomy
Topicssemigroups and automata theory · Finite Group Theory Research · Coding theory and cryptography
