Timeline for Existence of a finitely presented simple group satisfying certain properties
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 21 at 20:16 | comment | added | ADL | Are you aware of Osin's construction of groups with prescribed conjugacy classes? It is on the arXiv here. This can be applied to give finitely generated simple groups which contain each of the subgroups you wish, and such that all elements of the same order are conjugate. (So misses on a few counts, but suggests hope...) I suspect promoting the finite generation to finite presentability in his construction would be extremely difficult though. | |
| Jan 21 at 19:38 | comment | added | Daniel Sebald | $D_6$ and $A_4$ are the smallest pair of nonisomorphic non-Abelian groups with isomorphic Sylow subgroups. | |
| Jan 21 at 17:36 | comment | added | Moishe Kohan | What is the motivation for this question? I would have understood if you asked in general about torsion in fp simple groups... | |
| Jan 21 at 15:00 | comment | added | Matt Zaremsky | The only f.p. simple groups I know about are either torsion-free (Burger-Mozes, Hyde-Lodha), contain unbounded torsion (Higman-Thompson, Roever-Nekrashevych, Brin-Thompson, etc etc), or have infinitely many torsion conjugacy classes (Caprace-Remy following Kac-Moody). So we don't even have an example with exactly $n$ conjugacy classes of finite subgroups for any $1<n<\infty$, much less with any prescribed isomorphism types. | |
| Jan 21 at 14:47 | comment | added | Satan's Minion | @Andy Putman: Ok, I deleted my comment. | |
| Jan 21 at 13:08 | comment | added | Geoff Robinson | Do you have any theoretical reason to believe there might? Or do you have a reason why you would like to see such a group to highlight some particular property? | |
| Jan 21 at 12:41 | comment | added | Andy Putman | @Satan'sMinion: While I agree that the question would benefit from some motivation, your comment is way out of line. | |
| Jan 21 at 6:27 | review | Close votes | |||
| Jan 26 at 3:03 | |||||
| Jan 21 at 5:39 | history | asked | Daniel Sebald | CC BY-SA 4.0 |