Timeline for Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Jan 26, 2017 at 13:03 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Updated links.
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| Jun 16, 2016 at 12:16 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Updated some links.
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| S Jun 15, 2014 at 14:30 | history | bounty ended | CommunityBot | ||
| S Jun 15, 2014 at 14:30 | history | notice removed | CommunityBot | ||
| Jun 13, 2014 at 20:15 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Added links to databases which contain more examples.
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| Jun 11, 2014 at 16:54 | comment | added | Stefan Kohl♦ | @Wolfgang: Order 24, both divide: $\tau_{0(3),11(18)} \cdot \tau_{3(4),9(28)}$ Orders 84, 168, 420: I don't know any where both moduli divide (but it is quite possible that, enlarging the search bounds, one would find some). Order 24, none divide: $\tau_{0(6),10(15)} \cdot \tau_{6(8),0(20)}$. Order 84, none divide: $\tau_{0(8),6(20)} \cdot \tau_{2(9),6(30)}$. Order 168, none divide: $\tau_{0(25),17(30)} \cdot \tau_{11(12),21(32)}$. Orders 120 and 420, none divide: I don't know any example so far. | |
| Jun 11, 2014 at 16:41 | comment | added | Wolfgang | sure. And I suppose you have noticed that in many (but not all, see the 120 one) of the examples for seldom orders, there is one pair where $m_1$ does not divide $m_2$ (or vice versa). No idea if that can tell us anything. Have you found examples also for some others of the seldom orders where for each pair one divides the other? and are there pairs of seldom order where in both factors the $m_i$ do not divide? | |
| Jun 11, 2014 at 15:59 | comment | added | Stefan Kohl♦ | @Wolfgang: I think it is not really just "a coincidence", but so far I haven't been able to use it to find pairs yielding further orders. In any case by far not all pairs with orders 24, 84, 168 or 420 have $(m_1,m_2) = (8,20)$ -- there are many others as well. -- E.g. $\tau_{3(4),13(24)} \cdot \tau_{1(25),19(30)}$: order 24, $\tau_{0(3),1(30)} \cdot \tau_{11(20),5(32)}$: order 84, $\tau_{1(5),4(30)} \cdot \tau_{10(12),20(32)}$: order 168, $\tau_{0(15),10(30)} \cdot \tau_{10(20),28(32)}$: order 420. | |
| Jun 11, 2014 at 15:14 | comment | added | Wolfgang | four of the "seldom" orders, 24, 84, 168, 420, are yielded by a pair with $(m_1,m_2) =(8,20)$. Is that a coincidence or rather just a pair you have chosen in your search? Maybe searching with (8,28) or (12,20) and some random numbers for the rest yields more orders? or (9,15)? just a suggestion into the blue... | |
| Jun 11, 2014 at 9:36 | history | edited | Peter McNamara | CC BY-SA 3.0 |
changed link to something more accessible
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| S Jun 7, 2014 at 12:38 | history | bounty started | Stefan Kohl♦ | ||
| S Jun 7, 2014 at 12:38 | history | notice added | Stefan Kohl♦ | Draw attention | |
| Dec 8, 2013 at 14:26 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Added a note that the question will appear in the forthcoming 18th Edition of the Kourovka Notebook, where I had submitted it to after it had remained unsolved here for some months.
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| May 10, 2013 at 12:21 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Added a list of examples.
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| May 6, 2013 at 19:52 | comment | added | Stefan Kohl♦ | @Gerhard: at least none that I know. | |
| May 6, 2013 at 18:42 | comment | added | Gerhard Paseman | Is there a nice expanation for why 5 is not seen as an order in small examples? Gerhard "Ask Me About System Design" Paseman, 2013.05.06 | |
| May 6, 2013 at 18:00 | comment | added | Stefan Kohl♦ | Examples with order 15 are e.g. $\tau_{0(3),2(3)} \cdot \tau_{0(2),1(4)}$, $\tau_{2(3),1(6)} \cdot \tau_{1(4),3(8)}$ and $\tau_{5(6),8(12)} \cdot \tau_{2(9),8(9)}$. | |
| May 6, 2013 at 17:41 | comment | added | Douglas Zare | Could you give some examples with order $15$? | |
| May 6, 2013 at 17:07 | history | asked | Stefan Kohl♦ | CC BY-SA 3.0 |