I've learned that all non-abelian finite simple groups are $2$-generated, i.e. have a generating set of cardinality $2$. Is there a reference to this statement which does not just point to the classification of finite simple groups in general?
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$\begingroup$ The statement you've written is incorrect for the standard definition of the word "rank". Perhaps you are thinking of the statement that all finite simple groups can be generated by $2$ elements? $\endgroup$– Nick GillOct 14, 2015 at 11:22
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$\begingroup$ Yes, by rank of a group I mean the smallest cardinality of a generating set of this group. $\endgroup$– Sergei NemirovOct 14, 2015 at 11:28
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1$\begingroup$ CFSG is necessary for this result. However, most cases (groups of Lie type) are covered by a single Theorem of R. Steinberg. Alternating groups are easily dealt with. Sporadic groups (by their nature) are done on an ad hoc basis. $\endgroup$– Geoff RobinsonOct 14, 2015 at 11:49
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$\begingroup$ My first comment was a bit off: you're right that the rank of a group has the definition that you give it. However, there are a bunch of other meanings for the word rank and (at least in my world), they tend to be used more (e.g. rank of a permutation group, rank of an algebraic group etc) so clarifying the definition is a good idea. In any case, as Geoff says, all known proofs of this result require CFSG. A proof without CFSG would be of great interest. (Note that with CFSG you can make even stronger statements; for instance, all finite simple groups are $\frac32$-generated.) $\endgroup$– Nick GillOct 14, 2015 at 12:29
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2$\begingroup$ Closely related question: mathoverflow.net/q/59213/10266 $\endgroup$– Frieder LadischOct 20, 2015 at 13:26
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1 Answer
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The proof of the stronger statement that two random elements generate with high probability was completed by Liebeck and Shalev in:
Liebeck, Martin W.(4-LNDIC); Shalev, Aner(IL-HEBR-IM)
The probability of generating a finite simple group. (English summary)
Geom. Dedicata 56 (1995), no. 1, 103–113.
20P05 (20D06 20E18 20G40)
The math review has extensive references to work leading up to this result.