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 23β-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
23β-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 23β-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
23β-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.Comment: 38 pages; survey article based on my lecture at Groups St Andrews
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