Embedding of groups and quadratic equations over groups

TL;DR

This paper demonstrates that for any integer n ≥ 2, groups can be embedded into 2-generated groups while preserving the solvability of quadratic equations of length at most n, bridging group embedding and quadratic equation solvability.

Contribution

It introduces a method to embed arbitrary groups into 2-generated groups without losing solutions to quadratic equations of bounded length.

Findings

Groups can be embedded into 2-generated groups preserving quadratic equation solvability.

The embedding maintains the solvability status of quadratic equations of length ≤ n.

The result applies to both finite and countable groups.

Abstract

We prove that, for every integer $n \ge 2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, i.e., every quadratic equation over $G$ of length at most $n$ has a solution in $G$ if and only if this equation, considered as an equation over $H$, has a solution in $H$.