Timeline for Maximal subgroups of simple groups with normal $2$-subgroups
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Feb 12, 2018 at 11:08 | vote | accept | spin | ||
| Feb 1, 2018 at 12:21 | comment | added | Geoff Robinson | Ok, sorry to query unnecessarily, my oversight. | |
| Feb 1, 2018 at 11:48 | comment | added | Derek Holt | Well ${\rm GL}(r,3)$ has an element of order $2$ in its centre, so the wreath product with $S_s$ contains an elementary abelian $2^s$ as normal subgroup. On passing to ${\rm PSL}(rs,3)$ we lose at most two of the factors from the $2^s$, so if $s \ge 3$ then there we have a nontrivial normal $2$-subgroup. | |
| Jan 31, 2018 at 19:53 | comment | added | spin | What originally brought me to wonder about this question was Iwasawa's simplicity criterion: if $G$ is a finite group such that $G = [G,G]$ and $S$ is a primitive $G$-set such that $\operatorname{Stab}_G(x)$ has a normal solvable subgroup whose $G$-conjugates generate $G$, then $G$ is simple. So one could ask if there is a converse for this. Then we are basically asking for $G$ simple whether there exists a maximal subgroup $M$ of $G$ with a nontrivial normal solvable (equivalently nontrivial normal abelian) subgroup. This weaker statement holds, as noted in the comments by Geoff Robinson. | |
| Jan 31, 2018 at 19:45 | comment | added | spin | A bit disappointing that the result is not true for all simple groups in the end, but thank you for the very nice answer. | |
| Jan 31, 2018 at 17:08 | comment | added | Nick Gill | Except that I'm slightly unsure whether Gorenstein's result is for maximality over all 2-locals, or over all subgroups -- if the latter, then it would be a nice companion to yours... But I suspect it's the former.... | |
| Jan 31, 2018 at 17:08 | comment | added | Nick Gill | +1 for a lovely answer. I am reminded of Gorenstein's result about maximal 2-local subgroups: If $G$ is a simple group with only one conjugacy class of maximal 2-local subgroups, then $G$ is isomorphic to $L_2(2^n)$, $Sz(2^n)$ or $U_3(2^n)$ for some $n\geq 2$. (in his paper "On finite simple groups of characteristic 2 type"). This seems like the "next" set of simple groups to consider after the ones you've classified... | |
| Jan 30, 2018 at 17:45 | comment | added | Geoff Robinson | Still a nice answer and a good question. | |
| Jan 30, 2018 at 14:01 | history | edited | Derek Holt | CC BY-SA 3.0 |
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| Jan 30, 2018 at 13:01 | history | edited | Derek Holt | CC BY-SA 3.0 |
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| Jan 30, 2018 at 12:55 | comment | added | Derek Holt | @GeoffRobinson unfortunately I failed to check the small exceptional cases correctly, and I found a mistake. But I guess the corrected answer is also interesting. | |
| Jan 30, 2018 at 12:54 | history | edited | Derek Holt | CC BY-SA 3.0 |
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| Jan 30, 2018 at 11:11 | comment | added | Geoff Robinson | Very nice answer. This is indeed a surprising outcome, which I for one initially would not have expected, and which makes the question a really good one. I suppose in another sense it indicates that there are maximal subgroups of some simple groups which don't reflect very much about the group itself. | |
| Jan 29, 2018 at 20:10 | history | edited | Derek Holt | CC BY-SA 3.0 |
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| Jan 29, 2018 at 16:50 | history | answered | Derek Holt | CC BY-SA 3.0 |