Timeline for A question about formulating first order axioms for group theory.
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21, 2012 at 18:42 | vote | accept | Garabed Gulbenkian | ||
| Oct 21, 2012 at 18:41 | vote | accept | Garabed Gulbenkian | ||
| Oct 21, 2012 at 18:42 | |||||
| Oct 21, 2012 at 14:40 | comment | added | Dave Marker | The ultraproduct of non-abelian simple groups need not be simple. This is discussed in an earlier mathoveflow question: mathoverflow.net/questions/32908/… | |
| Oct 21, 2012 at 0:26 | comment | added | Andreas Blass | I was taking "simple" to include "non-abelian", as group-theorists sometimes do, but as I probably shouldn't. | |
| Oct 20, 2012 at 23:33 | comment | added | Noah Schweber | Specifically, it is infinite, so has proper non-trivial subgroups, and is abelian, so every subgroup is normal; but since each finite cyclic group of prime order is simple, if there were a first-order characterization of simplicity, the ultraproduct would have to also be simple, so we're done. (Or have I made a mistake somewhere?) | |
| Oct 20, 2012 at 22:56 | comment | added | Noah Schweber | In fact, isn't the ultraproduct of the finite cyclic groups of prime order not simple? | |
| Oct 20, 2012 at 22:55 | comment | added | Noah Schweber | The question of whether "sporadic" is first-order seems ill-formed to me, since I'm not sure there's a definition of sporadic in the first place, but towards showing the non-first-orderness of simplicity: is the ultraproduct of a sequence of increasingly large simple groups, by a non-principal ultrafilter, known to be simple? I would guess that it would generally not be simple, but I can't prove that. | |
| Oct 20, 2012 at 20:59 | history | answered | Andreas Blass | CC BY-SA 3.0 |