Timeline for Solvability of finite groups of order coprime to 15 -- proof without using CFSG?
Current License: CC BY-SA 3.0
12 events
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| Jan 24, 2020 at 13:52 | comment | added | Stefan Kohl♦ | @ThomasBrowning Essentially you are right -- just note that it was not a priori clear that every finite simple group indeed has a minimal simple subgroup. | |
| Jan 24, 2020 at 4:41 | comment | added | Thomas Browning | @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? | |
| Jul 16, 2013 at 23:22 | vote | accept | Stefan Kohl♦ | ||
| Jul 16, 2013 at 23:21 | answer | added | Stefan Kohl♦ | timeline score: 4 | |
| Jun 23, 2013 at 19:54 | comment | added | YCor | @Stefan: you're obviously right and I didn't say or suggest the contrary. But CFSG is made up of various sub-results, one of which is Feit-Thompson, which is certainly not the easiest. So it is natural to ask whether a given result makes use of it; more generally it is meaningful to wonder on which part of the classification does a given result rely on; obviously a general consequence of CFSG as you mention will not make use of the full classification. There are also statements not relying on any part of the classification, e.g. that for $G$ nonabelian simple we have $gcd(|g|,4)\neq 2$. | |
| Jun 23, 2013 at 9:55 | comment | added | Stefan Kohl♦ | @Derek: Maybe you could turn the nice answer from your comment into an answer which shows up as such? -- I think it is nicer if questions which are completely answered are not shown with '0 answers' ... . | |
| Jun 21, 2013 at 16:07 | comment | added | Stefan Kohl♦ | @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, 2013 at 15:15 | comment | added | YCor | 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? | |
| Jun 21, 2013 at 13:37 | comment | added | Stefan Kohl♦ | @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, 2013 at 10:35 | comment | added | Derek Holt | 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, 2013 at 10:29 | comment | added | user6976 | Can you use Thompson's N-groups en.wikipedia.org/wiki/N-group_(finite_group_theory) ? | |
| Jun 21, 2013 at 10:23 | history | asked | Stefan Kohl♦ | CC BY-SA 3.0 |