Timeline for Searching for theorems characterizing when $O_p(G)$ is trivial / non-trivial
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 23, 2020 at 1:47 | vote | accept | LSt | ||
| Aug 23, 2020 at 1:42 | comment | added | LSt | @Geoff Robinson and @ David A. Craven: Thank you very much. | |
| Jul 11, 2020 at 12:35 | comment | added | David A. Craven | True. I'm not sure that the odd $Z^*$ theorem is in many of them though. It won't be in Aschbacher, Gorenstein, Suzuki, etc. | |
| Jul 10, 2020 at 22:11 | comment | added | David A. Craven | I'm surprised you didn't mention the $Z_p^*$-theorem, @GeoffRobinson! Suppose that $O_{p'}(G)=1$. If $z$ is an element of order $p$ lying in a Sylow $p$-subgroup $P$ of $G$, and $z$ is not $G$-conjugate to any other elements of $P$, then $z$ lies in $Z(G)$. In particular $z\in O_p(G)$. For $p=2$ this is due to Glauberman, but for $p$ odd it only follows from CFSG. | |
| Jul 10, 2020 at 15:04 | comment | added | Geoff Robinson | Aschbacher's' book on Finite Groups would be one. | |
| Jul 10, 2020 at 12:47 | vote | accept | LSt | ||
| Aug 23, 2020 at 1:47 | |||||
| Jul 10, 2020 at 12:47 | comment | added | LSt | Ok, thank you. Yes, I mean the largest normal $p$-subgroup of $G$. Is there one (or more) particular group theory book which you would recommend (since a few don't treat $O_p(G)$) ? | |
| Jul 10, 2020 at 12:12 | history | answered | Geoff Robinson | CC BY-SA 4.0 |