Timeline for Conjugacy of nilpotent injectors in soluble groups
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Sep 7, 2023 at 23:02 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
corrected text
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| Sep 7, 2023 at 22:47 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
misleading word normal removed
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| Sep 7, 2020 at 15:58 | comment | added | Wembley Inter | @user13040 Thanks for the information! However, the proof of the general case given by Hans Lausch is based on knowing the maximal nilpotent groups containing Fitting subgroup are conjugate in advance. I wonder if you have already found the direct proof of the originial case. Thanks! | |
| May 22, 2011 at 2:51 | comment | added | user13040 | I contacted Isaacs himself about this and he said that his 'favorite proof' is something more general: if the intersection of two maximal nilpotent subgroups of a finite group G contains its own centralizer, then these subgroups are conjugate. I think the result Isaacs mentions is due to Hans Lausch in his paper "Conjugacy classes of maximal nilpotent subgroups" from 1984 springerlink.com/content/p106nl14504n8648. I would be in debt to anyone who might consider sending this paper to the following address: [email protected], as I'm having trouble accessing springer remotely. | |
| May 20, 2011 at 6:55 | comment | added | Geoff Robinson | Well, Jack Schmidt has now provided you with references in English. I have never read Fischer's original paper, but have cooked a proof by myself in the past, which I assume was the same as Fischer's. | |
| May 20, 2011 at 2:28 | comment | added | user13040 | Indeed it is not; I should have expressed myself in a better way. Regarding Fischer's proof, are you referring to "Klassen konjugierter Untergruppen in endlichen auflösbaren Gruppen"? If so, then the reason I didn't read it in the first place is because I don't know any german and was unable to find a 'reproduction' in english. To be honest I am out of my league here; I encountered this as a problem in Isaacs' 'Finite Group Theory' book (specifically 3C.8 in page 91) and, while mathoverflow is not meant to be a problem-solver, I thought it wouldn't be completely inappropriate to ask. | |
| May 20, 2011 at 0:15 | comment | added | Geoff Robinson | In that case, you are really looking for an elementary proof of Fischer's theorem, as mentioned by Jack Schmidt. Well, as Jack mentions, Fischer gave this proof before Fitting classes were invented, so Fischer's own proof is elemntary in the sense you wanted. Once we know that these "Fischer subgroups" are unique up to conjugacy, they must be nilpotent injectors in the above sense. But then I am slightly perplexed by the question, since existence is not an issue. | |
| May 19, 2011 at 22:12 | comment | added | user13040 | Thank you for your answer. I take a nilpotent injector of a soluble group to be a nilpotent subgroup containing the Fitting subgroup and maximal (under inclusion) with that property. | |
| May 19, 2011 at 16:00 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
minor typos
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| May 19, 2011 at 15:44 | vote | accept | CommunityBot | ||
| May 19, 2011 at 14:48 | history | edited | Geoff Robinson | CC BY-SA 3.0 |
expanded text to clarify answer.
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| May 19, 2011 at 9:01 | history | answered | Geoff Robinson | CC BY-SA 3.0 |