Is the solvability of finite groups of order coprime to 15 essentially easier to prove than the entire Classification of Finite Simple Groups?

1$\begingroup$ Can you use Thompson's Ngroups en.wikipedia.org/wiki/Ngroup_(finite_group_theory) ? $\endgroup$– user6976Jun 21, 2013 at 10:29

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 preclassification. But I have a suspicion that neither of them was ever published! $\endgroup$– Derek HoltJun 21, 2013 at 10:35

$\begingroup$ @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 nonabelian simple group contains a subgroup which is a minimal simple group".) $\endgroup$– Stefan Kohl ♦Jun 21, 2013 at 13:37

1$\begingroup$ @Yves: The proof of FeitThompson's Odd Order Theorem has something like 200 pages, while even the secondgeneration 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 FeitThompson. $\endgroup$– Stefan Kohl ♦Jun 21, 2013 at 16:07

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 Ngroup result as Mark mentioned? $\endgroup$– Thomas BrowningJan 24, 2020 at 4:41
1 Answer
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 preclassification, though they might not have been published.

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

1$\begingroup$ Glauberman's classification of S_4free 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

$\begingroup$ @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

2$\begingroup$ No, I meant what I sai. I think Glauberman published an AMS Monograph on S_4free group. Clearly, an S_3free group is S_4 free, so Glauberman's S_4free classification includes the classification of S_3free groups. $\endgroup$ Oct 6, 2013 at 22:16