Generalized presentations of infinite groups, in particular of $Aut(F_{\omega})$

TL;DR

This paper introduces a new framework for generalized group presentations and applies it to symmetric groups and automorphism groups of free groups of countable rank, expanding understanding of their algebraic structures.

Contribution

It develops a theory of generalized presentations and provides explicit generalized presentations for $ ext{Sym}(X)$ and $Aut(F_{ ext{omega}})$, broadening the scope of group presentation methods.

Findings

Generalized presentations of $ ext{Sym}(X)$ for any set $X$

Generalized presentations of $Aut(F_{ ext{omega}})$

Enhanced understanding of automorphism groups of free groups

Abstract

We develop a theory of generalized presentations of groups. We give generalized presentations of the symmetric group $\Sigma(X)$ for an arbitrary set $X$ and of the automorphism group of the free group of countable rank, $Aut(F_{\omega})$.