Timeline for $G\cdot H$ with $G,H$ non-Abelian finite simple
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2023 at 9:16 | comment | added | YCor | The remaining details. Consider an extension $1\to G\to L\to H\to 1$. By the above solvability result, the natural homomorphism $H\to\mathrm{Out}(G)$ is trivial, so the natural homomorphism $L\to\mathrm{Aut}(G)$ maps into $\mathrm{Inn}(G)$. Hence $GC_L(G)=L$ ($C_L(-)$ denoting the centralizer). Since $G$ has a trivial center, $G\cap C_L(G)=\{1\}$. Hence $L=G\times C_L(G)$. | |
| Sep 1, 2023 at 20:44 | comment | added | Derek Holt | If the action of $H$ on $G$ was trivial then the group would be $G \times H$. | |
| Sep 1, 2023 at 19:46 | comment | added | Daniel Sebald | @DaveBenson so? Why can’t $H$ act trivially on $G$? | |
| Sep 1, 2023 at 19:32 | comment | added | Dave Benson | On the other hand, there is quite an interesting non-split extension of the shape $(A_5)^6.A_5$, where $A_5$ is the alternating group of degree five. | |
| Sep 1, 2023 at 19:14 | comment | added | Derek Holt | Of course that result depends on CFSG. | |
| Sep 1, 2023 at 19:13 | comment | added | Dave Benson | No. The outer automorphism group of every finite simple group is soluble. Of course, the proof involves the classification. | |
| Sep 1, 2023 at 19:09 | review | Low quality posts | |||
| Sep 1, 2023 at 21:52 | |||||
| Sep 1, 2023 at 18:44 | history | asked | Daniel Sebald | CC BY-SA 4.0 |