The theorems on freedom for relatively free Lie algebras and groups with a single relation

TL;DR

This paper proves theorems demonstrating the conditions under which relatively free Lie algebras and groups with a single relation are free, extending classical results by Magnus and Shirshov.

Contribution

It establishes new theorems on the freedom of relatively free Lie algebras and groups with one relation, generalizing known classical results.

Findings

Proves the theorem on freedom for relatively free groups with a single relation

Proves the theorem on freedom for relatively free Lie algebras with a single relation

Extends classical results of Magnus and Shirshov

Abstract

In this paper we prove the theorem on freedom for relatively free groups with a single relation (analogous with the well-known result of Magnus) and the theorem on freedom for relatively free Lie algebras with a single relation (analogous with the well-known result of Shirshov).