Timeline for Simplicity of alternating group $A_n$
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 27, 2020 at 8:19 | history | edited | YCor |
edited tags; edited tags
|
|
| Apr 27, 2020 at 7:50 | history | edited | YCor | CC BY-SA 4.0 |
removed period (added alternating) from title (the question was bumped anyway)
|
| Sep 21, 2018 at 13:02 | vote | accept | Igor Rivin | ||
| Mar 28, 2016 at 12:43 | comment | added | alpoge | Incidentally for generals prep I'd wondered about this, but I don't think this is at all what you want: by Murnaghan-Nakayama, the only normal subgroups of S_n, n\geq 5, are the evident ones. Hence if H in A_n is normal it suff. to show that H\cap tHt^{-1} is nontrivial (t a transp.). Thus it suff. to find an element of H in a conj. class that doesn't split --- i.e. isn't a product of odd cycles of distinct lengths. Taking any such element, we're done (by taking powers) unless wlog it's (123...p), p\neq 3. Multiplying it with its conjugate under (12)(34) we get a (p-2)-cycle, QED by induction. | |
| Mar 27, 2016 at 15:17 | comment | added | Johannes Hahn | @NickGill The Iwasawa argument works for the orthogonals work too. One can use the same action for all the other classical groups: Act on the subset of the projective space consisting of isotropic points. This is (most of the time) a rank-3-action and primitive. | |
| Mar 27, 2016 at 12:26 | answer | added | Jan-Christoph Schlage-Puchta | timeline score: 8 | |
| Mar 9, 2016 at 20:52 | comment | added | Igor Rivin | @NickGill Keith Conrad's list seems to miss the simplest argument (the one in Milne's notes...) | |
| Mar 9, 2016 at 16:18 | answer | added | Geoff Robinson | timeline score: 11 | |
| Mar 9, 2016 at 15:55 | answer | added | Nick Gill | timeline score: 22 | |
| Mar 9, 2016 at 15:05 | review | Close votes | |||
| Mar 10, 2016 at 0:48 | |||||
| Mar 9, 2016 at 14:56 | comment | added | YCor | For small values, there are several MathStackExchange posts (in my opinion the question belong on there) $A_5$: math.stackexchange.com/questions/328870 math.stackexchange.com/questions/1540721 math.stackexchange.com/questions/1016101 and for $A_6$: math.stackexchange.com/questions/1068051 | |
| Mar 9, 2016 at 14:54 | comment | added | Nick Gill | This document has FIVE proofs of the simplicity of $A_n$, for $n\geq 5$: math.uconn.edu/~kconrad/blurbs/grouptheory/Ansimple.pdf | |
| Mar 9, 2016 at 14:23 | comment | added | Nick Gill | ... Iwasawa's Criterion will also do all the other alternating groups if you consider the action on 3-subsets. (Actually, rather than say all the classicals, perhaps I should hedge my bets and exclude the orthogonals -- I can't immediately remember the way the proof goes there.) I wrote a bunch of notes on this by the way that I am happy to send you (although it's all very classical). | |
| Mar 9, 2016 at 14:23 | comment | added | Dima Pasechnik | compute the character table, and see that only the trivial character has a kernel. | |
| Mar 9, 2016 at 14:07 | comment | added | Nick Gill | Using Iwasawa's Criterion would be a good thing to do if you're thinking of generalizing to other FSG's (it's good for all the classicals). | |
| Mar 9, 2016 at 14:02 | history | edited | Igor Rivin | CC BY-SA 3.0 |
fixed typos
|
| Mar 9, 2016 at 13:56 | history | asked | Igor Rivin | CC BY-SA 3.0 |