All finite simple groups are generated by two elements, though apparently the proof is tens of thousands of pages long. Here’s an obvious follow-up:
What’s a short finite presentation of a simple group with 3 generators (and not 2)?
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$\begingroup$ One natural question is to find a f.g. simple group generated by 3 but not 2 elements, another is to fine a finitely presentable one. But I don't immediately see the point of asking a "short finite presentation", especially to start with. $\endgroup$– YCorCommented 5 hours ago
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2$\begingroup$ I don't know of a finitely presented (infinite) simple group that is definitively known to not be 2-generated, but there is a finitely generated simple group that is definitively not 2-generated (link.springer.com/article/10.1007/BF00969195) and a finitely presented (5 generators and 10 relations, so, pretty short) simple group in Corollary IV(ii) of (arxiv.org/pdf/1712.01091) that maybe someone who's good at computational manipulations could check whether or not it's 2-generated. $\endgroup$– Matt ZaremskyCommented 4 hours ago
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1$\begingroup$ @MattZaremsky's references: Guba - A finitely generated simple group with free 2-generated subgroups and Radu - New simple lattices in products of trees and their projections. $\endgroup$– LSpiceCommented 3 hours ago
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