# How did “Ore's Conjecture” become a conjecture?

The narrow question here concerns the history of one development in group theory, but the broader context involves the sometimes loose use of the term "conjecture". This goes back to older work of Oystein Ore, with whom I had only a slight interaction in 1963 as an entering graduate student at Yale (he generously found my reading knowledge of mathematical German adequate).

In 1951 his short paper Some remarks on commutators appeared in Proc. Amer. Math. Soc. here. He raised the question of expressing every element of a given group as a commutator, noting at the outset (but without a specific example): "In a group the product of two commutators need not be a commutator, consequently the commutator group of a given group cannot be defined as the set of all commutators,but only as the group generated by these." In the paper he proves that every element of a (nonabelian) finite or infinite symmetric group is in fact a commutator, while each element of an alternating group is a commutator in the ambient symmetric group. He notes that for the simple groups $A_n$ ($n \geq 5$) the proof can be adapted to show that each element is a commutator within the group, adding: "It is possible that a similar theorem holds for any simple group of finite order, but it seems that at present we do not have the necessary methods to investigate the question."

There was a lot of subsequent progress stimulated in part by work on the classification of finite simple groups, culminating recently in a definitive treatment: Liebeck-O’Brien-Shalev-Tiep, The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939–1008. (See a related discussion in an older MO question 44269.) But there is some distance between Ore's remark and the designation "conjecture", so I'm left with my question:

How did "Ore's Conjecture" become a conjecture?

This is not an isolated instance in mathematics, where terms such as "problem", "question", "(working) hypothesis" tend to morph into "conjecture". (This has happened to me, which is of course fine with me when my tentative suggestion turns out to be true.) An example I encountered as a reviewer occurs in a 1985 Russian paper by D.I. Panyushev with a title translated into English as A question concerning Steinberg's conjecture. The Russian word resembling "hypothesis" is also used in the sense of "conjecture", as it seems to be here; but Panyushev is providing counterexamples. As I noted in my review, Steinberg was explicitly stating a "problem" in his ICM address, not a conjecture, though he probably hoped for a positive solution.

ADDED: Igor's comments prompt me to mention that another 1951 paper dealt more directly than Ore's with commutators in the finite simple alternating groups: Noboru Ito, A theorem on the alternating group $\mathfrak{A}_n (n \geq 5)$. Math. Japonicae 2 (1951), 59–60. Both papers were reviewed by Graham Higman, who added a reference "Cf. O. Ore ...." to his one-line summary of Ito's paper. Since Ito went on to make substantial contributions to finite group theory, maybe one should refer to "Ito's conjecture" (even if he didn't formulate it any more explicitly than Ore did).

AFTERWORD: I asked the question partly out of curiosity after revisiting Ore's old paper, hoping there might be some further indication in the literature or at least in individual recollections of how to justify the transition to "conjecture". One person I should have asked years ago was one of my teachers Walter Feit, who joined the Yale faculty around 1962 and brought finite group theory into the department during the later years of Ore's career. Ore himself was never much involved with finite simple groups and in that period concentrated on graph theory. His speculative remark isn't at all strong, and he makes no mention of other simple groups (like the Mathieu groups or finite classical groups) known at the time. In the absence of evidence, I'm more inclined than Igor is to withhold judgment. The literature suggests a rather casual decision by other people to refer to "Ore's conjecture". Fortunately for his reputation it eventually turned out to be true, but this couldn't have been foreseen by him (or others) around 1950.

My other reason for asking for question is to emphasize that there is still no conjectural classification of which finite nonabelian groups consist of commutators and which don't. Everything known so far amounts to study of individual groups. In particular, why does simplicity contribute to a positive answer?

• In my experience, the Russian word resembling "hypothesis" ONLY means conjecture... – Igor Rivin Oct 6 '11 at 20:37
• It is marketing at work. In a grant proposal or report, "proving a conjecture" looks much more solid than "answering a question" or "solving a problem". (This is definitely so in Russian, and I believe English is not much different in this respect.) – Sergei Ivanov Oct 6 '11 at 21:59
• I agree with Sergei, and would add that "proving a conjecture" has more cachet than even "disproving a conjecture." A friend of mine once proved a very nice theorem that disproved a conjecture of a very famous mathematician. One of his NSF grant proposal reviewers commented that his work would have been much more interesting if he had proved the conjecture instead. Say what? It seems unfair that my friend should be penalized because the famous mathematician guessed wrong, but that is how the world seems to work. – Timothy Chow Oct 6 '11 at 22:55
• One of the intriguing parts of that Lohwater dictionary is the comment in the preface that "a few spurious words have been added as a trap for plagiarists". I'd love to know just one such fake word they put in the dictionary for that reason. – KConrad Oct 7 '11 at 4:53
• On genders: norma is Latin and feminine all right, but schema is neither: it's a Greek neuter (pl. is schemata, remember). In the case of hypothesis, that's a Greek feminine. – Lubin Oct 10 '11 at 5:19

1. Ore's question became a conjecture quite early (there are some papers published in Chinese(!) in the early sixties where the math review refers to the "Ore conjecture"). My conjecture is that Ore (who was a very well-known mathematician then, and a professor at Yale, reasonably central then as now) must have talked about this result, and conjectured it with somewhat greater conviction than in his paper.

2. @Gil confounds two different Thompsons -- Robert C (who proved some results on linear groups) and John G the Fields medalist who made the "Thompson conjecture".

3. In fact, most of the statement @Gil attributes to RC Thompson goes back to Shoda, who showed that in a complex semi-simple lie group every element is a commutator; this was extended by various people.

4. It is a conjecture I attribute to myself, but probably goes back to the ancients, that in every reasonable simple group every element is a commutator. In particular, this ought to be true in $Homeo(M^n)$ and in $Diffeo(M^n).$ For homeo and diffeo it is known that everything is a product of "few" commutators (I think from some early work of Armstrong you can get six for Diffeo; von Neumann and Ulam claimed something like four for $Homeo(S^2).$ You can look at my paper with Manfred Droste ("on extension of coverings") for more examples of when this commutator statement is known to be true, and why one might care.

5. The reason I mention the stuff above is that the question was probably going around already in the late forties, and Ore, very likely, decided to show it in a relatively simple interesting case (most of his paper is actually on the infinite symmetric group) and made a conservative conjecture (being a conservative guy, I guess), but probably he already guessed something like what I am saying above.

• Very interesting!! – Gil Kalai Oct 9 '11 at 15:36
• @Igor: Your comments are certainly interesting, though some of what's suggested in 1 and 5 is highly speculative and impossible to document one way or the other. Ore worked in various algebraic areas, though only tangentially in group theory; most of his later work involved graph theory, as well as some universal algebra. The early Chinese papers which apparently refer (for convenience?) to "Ore conjecture" don't seem to have any clear reason for doing so. I was looking for a straightforward paper trail, but maybe that just doesn't exist. – Jim Humphreys Oct 9 '11 at 20:46
• @Jim: I certainly could not find a "paper trail", but by the mid-sixties "Ore conjecture" seems to have been well entrenched. All the more surprising, since the Ore theorem (that every element of $A_n$ is a commutator) was apparently unknown at Harvard at the Husemoller was working on his thesis (late fifties, I am guessing), and Andy Gleason was called upon to produce a 1-line proof of a somewhat stronger result (in fairness to Ore, most of Ore's paper is on the infinite case, which does not follow from Gleason's argument.) – Igor Rivin Oct 9 '11 at 20:54
• @Igor Rivin: Regarding point 3, Shoda only worked over characteristic $0$ fields. The characteristic $p>0$ case is much more difficult when $p$ is small, and R. C. Thompson's contribution goes well beyond the work of Shoda. As far as I know, even now after the proof of the Ore conjecture, there is no other proof for the case $\mathrm{PSL}_n(\mathbb{F}_q)$ (all primes) than the one of Thompson. – A Stasinski Aug 31 '17 at 10:08

Two remarks: First, even when mathematicians do make a conjecture, there are various possible interpretations of what it means. In fact, this is the case even for different conjectures made by the same mathematicians. In this letter Shmuel Weinberger described five different meanings for what "a conjecture" is, referring to different conjectures he had made. (Here are two: "On other times, I have conjectured to lay down the gauntlet: “See, you can’t even disprove this ridiculous idea.” On yet other times, the conjectures come from daydreams: it would be so nice if this were true.")

Second, it is often the case that an affermative answer to a problem made by X is referred to "X's conjecture". Often this is justifiable, and reflects the real intention of X. (But sometimes, it causes some problems.) For example, what was referred to as "Borsuk's conjecture" (at least from the late 50s) was a positive answer (which was expected by Borsuk) to a problem posed by Borsuk in a paper from 1933.

I don't have an answer, but another example. Number theorists will be familiar with "Lehmer's conjecture," but Lehmer made no conjecture; he confined himself to reporting the facts as he knew them. Almost 50 years later I was at a conference where he said that in those days you had to have a lot more evidence than he had to call something a conjecture. I'd be curious to know when Lehmer's problem started to be called Lehmer's conjecture.

• Your question is a good one (probably worth posting separately), but I tried to avoid making my question too open-ended or community-wiki style. – Jim Humphreys Oct 6 '11 at 22:53
• It is still often called "Lehmer's question", but after a lot of computational as well as some theoretical results it has naturally turned into a conjecture. One can, of course, call it "Lehmer's question which we now treat as a conjecture". This would be more precise but is a bit too long. – Misha Belolipetsky Oct 7 '11 at 3:34
• I was about to post precisely this! Extra confusion is caused by there being another number-theoretic problem sometimes descibed as Lehmer's conjecture - the assertion that Ramanujan's tau function is nonzero - for which it's also not clear that he actually made the conjecture, rather than just suggesting the question. – Gray Taylor Oct 7 '11 at 16:48
• While this is interesting, I'm down-voting on this occasion since I think answers like this are getting off-track (see Jim Humphreys's comment, and my remarks on Valerio Capraro's answer). – Yemon Choi Oct 8 '11 at 22:05
• @Yemon, Jim's first sentence refers to a "narrow question" and a "broader context." I was responding to the broader context. As you know, it is common practice in Mathematics, when you can't answer a question, to answer (or ask) some related question instead, in the hope that the roundabout approach eventually comes back to shed light on the original question. – Gerry Myerson Oct 9 '11 at 10:28

Another instance of a remark that is now a popular conjecture might be the following:

In 1976, Alain Connes wrote: Apparently such an embedding ought to exist for all $II_1$-factor because it does for the regular representation of free groups (Connes, A. Classification of injective factors, Ann. of Math. 104 (1976) p. 105).

Now this is the (still unsolved) Connes embedding conjecture.

Maybe the difference between remark and conjecture is simply given by how much the author is sure of what (s)he is saying. What I mean is the following: Ore used the words ''It is possible'', that means that he was not quite sure; Connes used the words ''Apparently [...] ought to exist'', which means that he was quite sure. So in this latter case, the designation conjecture might be more appropriate.

• As I remarked to Gerry, I'm actually interested in the special case I outlined rather than in compiling further examples (interesting as they are). The "Ore conjecture" seems to be an extreme instance of elevating a remark to a conjecture, especially because Ore's remark comes from an era when there was no conjectured list of finite simple groups. – Jim Humphreys Oct 7 '11 at 19:11
• When you say that some thing "ought to exist" it is fairly reasonable to attribute to you the conjecture that it exists. – Gil Kalai Oct 8 '11 at 19:13
• @Gil: Indeed at the very end of my answer I told exactly that. @Jim: it seems to me that when a conjecture was actually a remark, the designation conjecture got used only because it gives some importance. I think that is normal that people want that their own work is considered important. And working on a ''remark'' is a bit different that working on a ''conjecture''. Therefore, I think that the elevation of a remark to a conjecture is merely by ''convenience'', in order to self-justify their own work. – Valerio Capraro Oct 8 '11 at 21:03
• By the way, in the few talks where I've seen it mentioned, it is the "Connes embedding problem", not the "Connes embedding conjecture". Same goes for the "free group factor problem", as far as I'm aware. – Yemon Choi Oct 8 '11 at 22:04
• Both the designation are used (see arxiv.org/find) (I agree that "problem" is more used than "conjecture") About the free groups, as far as I know, nobody at some point was convinced that the free group factors was isomorphic or not. So there is no name associated. Indeed a necessary condition to become a conjectue is that there is a name of a mathematician associated to the conjecture. – Valerio Capraro Oct 9 '11 at 11:03