Timeline for Periodic Automorphism Towers
Current License: CC BY-SA 2.5
24 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2016 at 16:24 | comment | added | Justin Benfield | As an interesting side note: I spent some time looking into a generalization of this, where I allowed any combination of $Aut$s, $Out$s, and $Inn$s to be applied to the given group, and for all groups of order less than 16, those trees are finite. As an example, I also created an image of the tree for SmallGroup(16,2) via Graphviz and uploaded it to google, it can be found here: drive.google.com/file/d/0BzoGc-Cf4OECdThIWFRIVHJjRHc/… | |
| Dec 12, 2015 at 3:38 | answer | added | Justin Benfield | timeline score: 8 | |
| Dec 9, 2015 at 9:02 | history | edited | YCor |
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| Dec 9, 2015 at 8:55 | answer | added | Justin Benfield | timeline score: 3 | |
| Jun 14, 2010 at 6:46 | comment | added | Simon Thomas | @Victor: No problem! | |
| Jun 12, 2010 at 21:31 | comment | added | Victor Protsak | @Simon: I am sorry if I misunderstood what you had meant about the status of the question. Thank you for the clarification. Please, roll back the tag if inappropriate. | |
| Jun 12, 2010 at 17:16 | history | edited | Simon Thomas | CC BY-SA 2.5 |
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| Jun 12, 2010 at 17:12 | comment | added | Simon Thomas | @Victor: I think it is more accurate to say that it is an open question whether or not my question is an open question. If you consider the statement of Scott's question, it suggests that he knew that there were finite groups such that the isomorphism type of $Aut^{(n)}(G)$ isn't eventually constant, but didn't know whether or not it was always eventually periodic. So perhaps the answer to my question was known formerly but has been forgotten? | |
| Jun 12, 2010 at 9:02 | answer | added | Guntram | timeline score: 3 | |
| Jun 12, 2010 at 7:38 | history | edited | Victor Protsak |
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| Jun 12, 2010 at 0:55 | comment | added | Simon Thomas | That result is already in my (unpublished) book (Theorem 5.2.9): here the real problem is whether you can manage a "triple jump". | |
| Jun 12, 2010 at 0:28 | comment | added | Joel David Hamkins | On the opposite side of period 2 for infinite groups: Gilbert Baumslag once told me an example of an infinite $G$ such that Aut(G) and Aut(Aut(G)) both jump maximally in cardinality, which as you know is impossible for centerless G, since this is how you got the bound on the tower height for centerless G. | |
| Jun 12, 2010 at 0:27 | history | edited | Simon Thomas | CC BY-SA 2.5 |
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| Jun 12, 2010 at 0:17 | history | edited | Simon Thomas | CC BY-SA 2.5 |
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| Jun 12, 2010 at 0:12 | comment | added | Simon Thomas | Mmmm ... maybe the infinite case is worth thinking about! | |
| Jun 12, 2010 at 0:11 | comment | added | Joel David Hamkins | Simon, yes, that's right. Although Scott inquired only about finite groups, as you say, the question could equally be asked about arbitrary groups (even non-centerless), and to my knowledge, no instances of this are known. (And I have asked many people.) I have long thought that this question for small finite groups might be amenable to computer search, but this has not been tried yet to my knowledge, despite my efforts to interest the people who could carry such an effort out. After all, there may be a very small period 2 example! | |
| Jun 11, 2010 at 23:30 | comment | added | Simon Thomas | @Joel: I have consulted your reference and it seems that the author doesn't even know the answer to my question when $G$ is an arbitrary group. Or have I forgotten something? | |
| Jun 11, 2010 at 23:04 | comment | added | Simon Thomas | @Kevin: because they don't answer the question. | |
| Jun 11, 2010 at 21:26 | comment | added | Joel David Hamkins | For everyone's information, some of the best answers to the other question amount to providing links to Simon Thomas' articles and book on the subject. | |
| Jun 11, 2010 at 21:21 | comment | added | Kevin O'Bryant | How are the answers for 5635 not adequate? | |
| Jun 11, 2010 at 20:30 | comment | added | Simon Thomas | @David: I hadn't noticed that this question had already been asked in mathoverflow.net/questions/5635/does-autaut-autg-stabilize. But it's a good question so it's worth asking again! | |
| Jun 11, 2010 at 20:10 | comment | added | David E Speyer | You might want to look at the answers to mathoverflow.net/questions/5635/does-autaut-autg-stabilize | |
| Jun 11, 2010 at 19:56 | history | edited | Simon Thomas | CC BY-SA 2.5 |
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| Jun 11, 2010 at 19:51 | history | asked | Simon Thomas | CC BY-SA 2.5 |