Is the solvability of finite groups of order coprime to 15 essentially easier to prove than the entire Classification of Finite Simple Groups?
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asked Jun 21, 2013 at 10:23
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1$\begingroup$ Can you use Thompson's N-groups en.wikipedia.org/wiki/N-group_(finite_group_theory) ? $\endgroup$– user6976Jun 21, 2013 at 10:29
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7$\begingroup$ 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! $\endgroup$– Derek HoltJun 21, 2013 at 10:35
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$\begingroup$ @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".) $\endgroup$– Stefan Kohl ♦Jun 21, 2013 at 13:37
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1$\begingroup$ @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. $\endgroup$– Stefan Kohl ♦Jun 21, 2013 at 16:07
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1$\begingroup$ @StefanKohl Perhaps I am missing something but isn't a minimal counterexample to "Every finite group of order coprime to 15 is solvable" necessarily a minimal simple group? So doesn't this follow from Thompson's N-group result as Mark mentioned? $\endgroup$– Thomas BrowningJan 24, 2020 at 4:41
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1 Answer
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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.
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1$\begingroup$ 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. $\endgroup$– Stefan Kohl ♦Jul 16, 2013 at 23:27
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1$\begingroup$ 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. $\endgroup$ Oct 6, 2013 at 21:16
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$\begingroup$ @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? $\endgroup$– Stefan Kohl ♦Oct 6, 2013 at 21:21
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2$\begingroup$ 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. $\endgroup$ Oct 6, 2013 at 22:16