# Solvability of finite groups of order coprime to 15 -- proof without using CFSG?

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

• Can you use Thompson's N-groups en.wikipedia.org/wiki/N-group_(finite_group_theory) ?
– user6976
Jun 21 '13 at 10:29
• 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! Jun 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".) Jun 21 '13 at 13:37
• @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. Jun 21 '13 at 16:07
• @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? Jan 24 '20 at 4:41

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.
• 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. Oct 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? Oct 6 '13 at 21:21