Timeline for Hall $\pi$ subgroups that controls its own fusion
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2016 at 17:24 | comment | added | mesel | For example, If the group $G$ can be smaller enough by induction so that $H$ is nilpotent, Proof can be finished directly by transfer teory. | |
| Oct 20, 2016 at 17:17 | comment | added | mesel | I think induction may work in that manner. By the way, I take $H$ as hall $\pi$ subgroup. | |
| Oct 20, 2016 at 17:13 | comment | added | mesel | Induction on $G$. We can assume that $O_{\pi'}(G)=1$, Otherwise we are done by induction. Let $Q$ be a Hall $\pi'$ subgroup of $G$. Notice that for $p\in \pi$, If $O_p(G)\neq 1$ then $\overline Q$ is normal in $G/O_p(G)$. Thus, $G=N_G(Q)O_p(G)$. Hence if $O_r(G)\neq 1$ then we have also $G=N_G(Q)O_r(G)$ which leads $N_G(Q)=G$. Thus, we can assume that $F=F(G)=O_p(G)$. And By induction, it can be seen that $F$ is minamal normal subgroup and $N_G(Q)$ is a maximal subgroup of $G$. | |
| Oct 20, 2016 at 16:50 | comment | added | Geoff Robinson | In fact, the $G = S_{5},H = S_{4}$ example shows what can go wrong (although $G$ itself is not solvable). In that example, $G^{\prime } = A_{5}$ and $H^{\prime} = A_{4}.$ It is still true that $H^{\prime}$ is a Hall $\{2,3\}$-subgroup of $G^{\prime},$ but $H^{\prime}$ no longer controls the fusion of its elements in $G^{\prime},$ since an element of order $3$ in $H^{\prime}$ is conjugate to its inverse in $G^{\prime},$ but not in $H^{\prime}.$ | |
| Oct 20, 2016 at 9:59 | comment | added | mesel | Yes by saying controls its own fusion I mean controls the G-fusion of its elements. Thank you very much. I had thought that it has a proof with Transfer theory and may be induction on the order of G for solvable group. | |
| Oct 19, 2016 at 23:13 | comment | added | Geoff Robinson | By the way, I have interpreted "controls its own fusion" as "controls the $G$-fusion of its elements". There are other possible interpretations. | |
| Oct 19, 2016 at 19:31 | vote | accept | mesel | ||
| Oct 19, 2016 at 17:53 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
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| Oct 19, 2016 at 17:40 | comment | added | Geoff Robinson | You have the proof in the countererexample. I will include the proof in my answer now for the other part. | |
| Oct 19, 2016 at 17:37 | comment | added | mesel | Dear Geoff Robinson thank you for your information and counter example but I need reference or a proof. | |
| Oct 19, 2016 at 17:29 | history | answered | Geoff Robinson | CC BY-SA 3.0 |