The answer is yes, because Thompson proved that the only finite simple groups of order coprime to 3 are the Suzuki groups, and Glauberman later extended this to a classification of simple groups that do not have $S_3$ as a subgroup. Bothe of thsoe results are pre-classification. But I have a suspicion that neither of them was ever published!
– Derek HoltJun 21 '13 at 10:35

@Derek, Mark: Thank you very much! (The result would follow also from the classification of minimal simple groups (Thompson, 1968) if one knows that every nonabelian finite simple group contains a minimal simple group, but is there a proof of the latter which is not based on CFSG? -- Barry and Ward (dmle.cindoc.csic.es/pdf/…) write "It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group".)
– Stefan KohlJun 21 '13 at 13:37

But what is the meaning of "without CFSG"? For instance does it allow making use of Feit-Thompson? Does Thompson's result about groups of order prime to 3 use Feit-Thompson?
– YCorJun 21 '13 at 15:15

1

@Yves: The proof of Feit-Thompson's Odd Order Theorem has something like 200 pages, while even the second-generation classification proof will have about 5000 pages. So I'd say definitely something can still be essentially easier to prove than CFSG if the proof needs Feit-Thompson.
– Stefan KohlJun 21 '13 at 16:07

As Derek Holt has pointed out, the answer to the question is yes. --
Thompson proved that the only finite simple groups of order coprime to 3 are the
Suzuki groups, and Glauberman later extended this to a classification of
simple groups that do not have ${\rm S}_3$ as a subgroup.
Both of these results are pre-classification, though they might not have been published.

Taking into account the suggestions on Meta (meta.mathoverflow.net/questions/493/…), I have turned Derek's comment into an answer. This is mainly to make clear that the question has already been answered.
– Stefan KohlJul 16 '13 at 23:27

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Glauberman's classification of S_4-free groups was definitely published. Also, a simple group of minimal order subject to having order prime to 3 has every proper subgroup solvable, so is an $N$-group, and would appear in Thompson's (published) list of $N$-groups, so the answer is yes on the basis of published results.
– Geoff RobinsonOct 6 '13 at 21:16

@Geoff: Thanks for your remark! -- By the way, is it possible that you rather mean ${\rm S}_3$-free groups than ${\rm S}_4$-free ones?
– Stefan KohlOct 6 '13 at 21:21

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No, I meant what I sai. I think Glauberman published an AMS Monograph on S_4-free group. Clearly, an S_3-free group is S_4 free, so Glauberman's S_4-free classification includes the classification of S_3-free groups.
– Geoff RobinsonOct 6 '13 at 22:16