The spread of finite and infinite groups

TL;DR

This survey explores recent advances in the understanding of how finite and infinite groups, especially simple groups, can be generated by small sets, highlighting new classifications, properties, and open questions in the field.

Contribution

It reviews recent generalizations of the $rac{3}{2}$-generation property for finite simple groups and discusses the discovery that certain infinite simple groups are also $rac{3}{2}$-generated.

Findings

Finite simple groups have the $rac{3}{2}$-generation property.

Finite $rac{3}{2}$-generated groups have been classified.

Finitely presented Thompson groups are $rac{3}{2}$-generated.

Abstract

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is contained in a generating pair. Much more recently, this result has been generalised in three different directions, which form the basis of this survey article. First, we look at some stronger forms of $\frac{3}{2}$-generation that the finite simple groups satisfy, which are described in terms of spread and uniform domination. Next, we discuss the recent classification of the finite $\frac{3}{2}$-generated groups. Finally, we turn our attention to infinite groups, focusing on the recent discovery that the finitely presented simple groups of Thompson are also $\frac{3}{2}$-generated, as are many of their generalisations. Throughout the article we pose open questions in this area, and we highlight connections with other areas of group theory.