Timeline for Can ugly groups have derived length 3?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2015 at 21:34 | comment | added | Derek Holt | There are no further examples of order up to $2000$ except possibly for order $1536$. There are $408641062$ groups of order $1536$, so that will take longer to check! I would guess that there no examples of derived length $3$, but I could be wrong. | |
| Jun 20, 2015 at 21:06 | comment | added | moshe noiman | On Thu, Apr 19, 2012 at 10:53 AM, Derek Holt <[email protected]> wrote: > I did a check through the small groups database (in GAP and Magma) and found that two groups of order 768: > SmallGroup(768, 1085345) and > SmallGroup(768, 1085350) > have this property. They are the only examples of order up to 1000. Bothe of these examples have derived length 4. I think it would be difficult to find a metabelian example, but they might still exist! > Best regards, > Derek Holt. | |
| Jun 20, 2015 at 19:32 | comment | added | moshe noiman | The definitions of good and bad groups are motivation for the definition of an ugly group. Ugly groups are obstacles to goodness. | |
| Jun 19, 2015 at 18:13 | comment | added | YCor | You have a definition of ugly groups, and two questions about ugly groups. Before this there is a definition of bad/good groups, with no question about them. Why did you introduce the definition of good/bad groups with no question about them? | |
| Jun 19, 2015 at 12:52 | comment | added | Derek Holt | Do you have a reference for the construction of ugly groups by D Holt? (Was it Hardy who said that there was no place in this world for ugly mathematics?) – | |
| Jun 19, 2015 at 8:38 | comment | added | moshe noiman | 3) Wreath products of cyclic groups are good, by definition, so by Kaloujnine-Krasner all (finite solvable) groups are subgroups of good groups. So goodness is not quotient-closed. 4) nilpotent, more generally supersolvable, more generally Sylow tower groups, are all good. Can the class be made larger without destroying subgroup-closed? | |
| Jun 19, 2015 at 8:37 | comment | added | moshe noiman | 1) A bad group must involve an ugly group, as a quotient of some subgroup. 2) By a result of Curran, a centerless group G of derived length 2 splits over G', so groups of derived length 2 can not be ugly, and must in fact be good. On ther other hand, D Holt used GAP to find ugly groups of derived length 4. Thus the question about derived length 3. | |
| Jun 19, 2015 at 8:36 | history | asked | moshe noiman | CC BY-SA 3.0 |