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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?

Stefan Kohl
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Sergei Nemirov
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  • 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? – Nick Gill Oct 14 '15 at 11:22
  • Yes, by rank of a group I mean the smallest cardinality of a generating set of this group. – Sergei Nemirov Oct 14 '15 at 11:28
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    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. – Geoff Robinson Oct 14 '15 at 11:49
  • 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.) – Nick Gill Oct 14 '15 at 12:29
  • Since you want references, the $\frac32$-generation result that I mentioned can be found in Guralnick, Robert, Kantor, William, Probalistic generation of finite simple groups, J. Algebra 234 (2000), p. 743–792. The result was proved simultaneously by Stein, Alexander, $1\frac12$-generation of finite simple groups. Beiträge Algebra Geom. 39 (1998), no. 2, 349–358. – Nick Gill Oct 14 '15 at 13:03
  • @NickGill What do you regard as the standard definition of "rank"? I try to avoid using it because it is so overused, but in my experince the most common or default intended meaning is minimal size of a generating set (as in rank of free group or free abelian group, etc.) – Derek Holt Oct 15 '15 at 08:05
  • @DerekHolt, like I said above, the standard definition is the one the OP used so my original comment was wrong. Mea culpa! (Having said that, it does seem ridiculous that this single word is so overloaded with meanings, varying from context to context. Surely mathematicians could be a little more imaginative with terminology....) – Nick Gill Oct 15 '15 at 08:49
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    Closely related question: http://mathoverflow.net/q/59213/10266 – Frieder Ladisch Oct 20 '15 at 13:26

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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.

Frieder Ladisch
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Igor Rivin
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