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Here is a very natural question:

Q: Is it always possible to generate a finite simple group with only $2$ elements?

In all the examples that I can think of the answer is yes.

If the answer is positive, how does one prove it? Is it possible to prove it without using the classification of finite simple groups?

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    $\begingroup$ Upon seeing this question, I initially parsed it as an easy question about the cyclic group $C_2$. $\endgroup$
    – James Cranch
    Aug 26, 2012 at 8:38

6 Answers 6

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The answer to your question is yes. Moreover, if you pick two random elements from a finite simple group, then they generate the whole group with probability which tends to 1 as the size of the group grows. There are even stronger results in this direction, but I am not an expert in the subject so you will have to look for it yourself. You should look for papers by Liebek and Shalev, Lubotzky, Kantor, and there are others who I am not sure about now.

All of these results require the classification. There are very few results regarding finite simple groups which do not require the classification.

Edit: here is a link to a fairly old survey paper in the notices: http://www.ams.org/notices/200104/fea-shalev.pdf. There are many new developments in the last decade.

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    $\begingroup$ 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. $\endgroup$
    – Kevin O'Bryant
    Mar 23, 2011 at 2:37
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    $\begingroup$ I guess the classification of finite simple groups is among these few results :-) $\endgroup$
    – Torsten Ekedahl
    Mar 23, 2011 at 11:44
  • $\begingroup$ @Kevin: Can you maybe reveal who this Guru is? $\endgroup$
    – j.p.
    Mar 23, 2011 at 13:50
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    $\begingroup$ Indeed Group Theory is a cult and we have gurus. Moreover, we sometimes sacrifice mathematical virgins to the monster. $\endgroup$
    – Yiftach Barnea
    Mar 23, 2011 at 14:05
  • $\begingroup$ @jp: I don't know which is Kevin's guru, but if you want someone to blame, then I heard experts like G. Malle, G. Stroth and Bernd Fischer himself say that on different occasions. $\endgroup$
    – Johannes Hahn
    Apr 17, 2012 at 12:07
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Since I happen to know the OP is number-theoretically inclined, let me add the following remark:

For "most" finite simple groups $G$ it is indeed the case that $G = \langle x, y \rangle$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a quotient of the free product $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z} \cong \operatorname{PSL}_2(\mathbb{Z}) = \Gamma(1)$.

This has the following geometric consequence: there is some subgroup $\Gamma_G \subset \Gamma(1)$ such that $X_G = \Gamma_G \backslash \overline{\mathcal{H}}$ is a modular curve and $X_G \rightarrow X(1) \cong \mathbb{P}^1$ is a $G$-Galois branched covering. By taking $G$ to be something else than $\operatorname{PSL}_2(\mathbb{Z}/p\mathbb{Z})$ one sees that $\Gamma(1)$ admits many non-congruence subgroups. For instance, it is well-known (added: I should have said "a well-known theorem of J.G. Thompson") that one can take $G$ to be the Fischer-Griess Monster.

I don't want to make precise what I mean by "most". Note that there are infinitely many finite simple groups with order prime to $3$ (although one has to look fairly far down the list of all finite simple groups to see them: Suzuki groups), so I definitely do not mean "all but finitely many".

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    $\begingroup$ For a slightly smaller value of "most" you can even say that for most finite simple groups you can choose xy to have order 7, that is most large rank simple groups (over most fields) are Hurwitz Groups: en.wikipedia.org/wiki/… — There are definitely lots of groups where this doesn't work, but I find it surprising how often it does work. – Jack Schmidt 0 secs ago $\endgroup$
    – Jack Schmidt
    Mar 23, 2011 at 14:19
  • $\begingroup$ @Jack: my definition of "most" does not extend to the statement "Most finite simple groups are Hurwitz groups", I think. But I'll think further about this... $\endgroup$
    – Pete L. Clark
    Mar 23, 2011 at 14:38
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    $\begingroup$ Hi Pete, very cool result! $\endgroup$
    – Hugo Chapdelaine
    Mar 23, 2011 at 16:30
  • $\begingroup$ The symmetries of a hexagon $D_6$ is generated by $\langle R, T \rangle $ , where $R$ is a rotation by $60^\circ$ and $T$ is a reflection about one of its axes of symmetry. While the order of $T$ is $2$, the order of $R$ is $6$, not $3$. This contradicts your statement. For "most" finite simple groups G it is indeed the case that G=⟨x,y⟩ where x has order 2 and y has order 3. You say most, I can't think of any that pop out. $\endgroup$
    – john
    Oct 14, 2022 at 9:58
  • $\begingroup$ @john: I couldn't find a statement in my answer that your comment contradicts. To see a sense in which "most" finite groups are (2,3)-generated, see mathoverflow.net/questions/365374/…. $\endgroup$
    – Pete L. Clark
    Oct 17, 2022 at 6:39
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In addition to the two answers already given it might be worth to mention that the generating graph of a finite simple group has no isolated vertices: This means that for every nonidentity element $x\in G$, there is some other element $y$ such that $G=\langle x, y\rangle$. (The generating graph of a group has the nonidentity elements of $G$ as vertices, where to vertices are connected if they generate the group.) This is shown in

Guralnick, Robert, Kantor, William, Probalistic generation of finite simple groups, J. Algebra 234 (2000), p. 743–792. (MR1800754)

Recently, Breuer, Guralnick, Lucchini, Maróti and Nagy have shown that the generating graph of every "sufficiently large" finite simple group contains a Hamiltonian cycle. You might also look at the references given in their paper:

Breuer, T., Guralnick, R. M., Lucchini, A., Maróti, A., Nagy, G. P., Hamiltonian cycles in the generating graphs of finite groups, Bull. Lond. Math. Soc. 42 (2010), p. 621–633. (MR2669683)

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    $\begingroup$ Nice answer. One comment: A group $G$ that has the property that for any element $x$ in $G\setminus\{1\}$ there is an element $y$ such that $\langle x,y\rangle=G$ is said to be $\frac32$-generated. The fact that all finite simple groups are $\frac32$-generated was also proved by Stein (in addition to Guralnick & Kantor, as you mention in your answer). The relevant reference is: Stein, Alexander, $1\frac12$-generation of finite simple groups. Beiträge Algebra Geom. 39 (1998), no. 2, 349–358. $\endgroup$
    – Nick Gill
    Dec 11, 2014 at 22:28
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There is a paper in arxiv by Robert Guralnick and Gunter Malle that answers your question in a stronger way. Their aim is to prove existence of algebraic surfaces obtained in a specific way as a quotient of finite group actions on products of curves of genus > 1. They prove the existence of two conjugacy classes in a finite simple group with the property that picking one element each from these classes always generates the group.

Here is the link:

http://arxiv.org/abs/1009.6183

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Yes. See this.

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  • $\begingroup$ This link appears to be broken. After doing some digging on David Rusin's NIU page, it appears like this is the current version of the link: math.niu.edu/Papers/Rusin/known-math/98/2generators $\endgroup$
    – Neil Hoffman
    Jul 29, 2015 at 5:31
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    $\begingroup$ @NeilHoffman That link also appears to be broken, is there an updated version? $\endgroup$
    – Ali Caglayan
    Mar 14, 2017 at 21:46
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    $\begingroup$ To whoever downvoted this recently: if I could repair the link, I would. I have written twice to find out the fate of Dave Rusin's excellent pages, to no avail. $\endgroup$
    – Todd Trimble ♦
    Jul 26, 2017 at 23:40
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    $\begingroup$ Archived version available here: web.archive.org/web/20100704042457/http://www.math.niu.edu/… $\endgroup$
    – niemiro
    Dec 9, 2018 at 13:34
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Carlisle King recently posted an arXiv preprint which (claims to) show that every finite simple group is generated by an involution, together with another element of prime order. http://arxiv.org/abs/1603.04717

The paper uses the classification, as most of these do.

Elements of prime (or even prime power) order seem to be particularly easy to work with for a number of kinds of arguments. See e.g. this mathoverflow question.

Now, if you want a result that doesn't use the classification, Paul Flavell gave an elementary proof that every non-solvable group has 2 elements that generate a non-solvable group. See here.

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    $\begingroup$ This paper has now appeared in J. Algebra (MSN). $\endgroup$
    – LSpice
    Jul 25, 2017 at 13:23

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