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?

provedthe 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. $\endgroup$ – Timothy Chow Oct 6 '11 at 22:55