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It is well-known that one of the corollaries of the classification of finite simple groups (CFSG) is that every finite simple group can be generated by two elements. In a comment on an answer to an old question, Kevin O'Bryant mentioned that

A group theory guru once told me that this is almost equivalent to the classification, in that the classification could be tremendously simplified if this (every finite simple group can be generated by two elements) could be taken as a lemma.

How does this simplification go?

An ideal answer would of course be one that gives a full proof of the CFSG assuming the $2$-generation property as a hypothesis, but that seems like a lot to ask for (and probably still involves some tedious case-by-case analysis?), so perhaps more reasonable is to ask about the broad outline of such a simplification.

As a bonus question, is there anything specific about $2$-generation here, or would it suffice to know that there exists some $k \geq 2$ such that every finite simple group can be generated by $k$ elements in order to achieve (some) simplification of the proof of the CFSG?

Carl-Fredrik Nyberg Brodda
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    I know precisely nothing about this question. But I’ll note that when talking about something on the scale of CFSG, a “tremendous simplification” might still mean a couple of hundreds of pages. – Emil Jeřábek Jan 12 '24 at 10:59
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    "A couple of hundreds of pages" would sound to me a bit optimistic :) What about the Feit-Thompson (odd order) theorem? Does it have a $<100$-page proof assuming that every finite simple group is 2-generated? – YCor Jan 12 '24 at 11:02
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    Or perhaps Emil meant a reduction by a couple of hundred pages? – Derek Holt Jan 12 '24 at 11:15
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    @YCor "Simplified" doesn't necessarily mean "fewer pages." Here's how Wikipedia describes one of the "simplified proofs" of F-T: "The simplified proof has been published in two books: (Bender & Glauberman 1994), which covers everything except the character theory, and (Peterfalvi 2000, part I), which covers the character theory. This revised proof is still very hard, and is longer than the original proof, but is written in a more leisurely style" (emphasis mine). – Timothy Chow Jan 12 '24 at 13:27
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    There is now a relatively short proof by P.J. Flavell of the fact that every minimal simple group is two-generated ( it does use methods of Thompson, Bender and Glauberman). J.G. Thompson's original proof of this fact was as a corollary of the enormous N-group paper, which served in many ways as a template for the classification. As far as I know, Flavell's proof has noy yet led to any significant simplification of the determination of the minimal simple groups. – Geoff Robinson Jan 29 '24 at 14:36
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    @YCor: In fact, in the case of a (supposed) finite non-Abelian simple group $G$ of odd order with all proper subgroups solvable, it follows from the Theorem of P. Flavell I mentioned above, that $G$ is two-generated. As far as I know, this fact has not yet led to significant simplifications in the proof of the odd order theorem. – Geoff Robinson Jan 29 '24 at 14:39

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I don't know who the "guru" being alluded to might have been. One obvious consequence of every finite non-Abelian simple group being two-generated is that for each finite simple group $G$, the outer automorphism group of $G$ has order less than $|G|$ (just consider where the pair of generators can be sent under an automorphism, and observe that $|{\rm Aut}(G)| \leq |G|(|G|-1).$

A priori, without knowing the precise structure of a putative finite simple group $G$ , we would know almost nothing about its outer automorphism group, and in general, there seems to be no real reason to expect a bound of much less than $|G|^{\log_{2}(|G|)}$ for the order of its automorphism group.

Since analysis of a maximal subgroup $M$ of a putative finite simple group $G$ often requires understanding of the outer automorphism group of the generalized Fitting subgroup $F^{\ast}(M)$, and since $F^{\ast}(M)$ is a central product of the nilpotent subgroup $F(M)$ with the quasisimple subnormal subgroups of $M$, any knowledge of the outer automorphism groups of quasi-simple groups is useful (and the outer automorphism group of a quasisimple group $L$ is the same as the outer automorphism group of its simple quotient $L/Z(L)$).

This is very far from a justification of the fact that there would be a dramatic simplification of the proof of CFSG, and I clearly can't know what that person had in mind (perhaps it was something deeper and much more subtle), but it does at least point towards some potential simplifications if the two-generation of simple groups could be taken for granted. However, I find it hard to imagine how it would be possible to take for granted that simple groups are two-generated without knowing an awful lot about them.

Geoff Robinson
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  • In response to the final paragraph here, I interpreted the original statement given by the "guru" not as a suggestion that there would be a "dramatic simplification" to the overall list of proofs, but rather that showing directly simple groups are 2-generated was likely to be as hard as the full proof of CFSG. Meaning: I suspect the guru is in complete agreement with this post (as indeed I myself am). – John McVey Mar 21 '24 at 18:33
  • Okay, I translated "tremendously simplified" to "a dramatic simplification" , but the sense of those expressions is much the same. I intended no criticism of anyone, I genuinely did not (and could not) know what the person might have had in mind. – Geoff Robinson Mar 21 '24 at 18:59
  • Understood, and I tried to word my comment not to intimate otherwise. I happen to come from a cryptography perspective, where "if A is easy, then factoring is easy" is actually a commitment that A is in fact hard. – John McVey Mar 21 '24 at 19:07
  • I understand the implication that the fact if it was easy to prove two generation of simple groups it would be easier to do CFSG means that proving two generation is hard. But the various statements have different nuances of meaning. A question of semantics/interpretation. – Geoff Robinson Mar 21 '24 at 19:12