Timeline for Locally finite compact groups
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 27, 2014 at 7:25 | comment | added | Benjamin Steinberg | @YvesCornulier, this is why I said essentially a duplicate because torsion profinite is locally finite and then the questions merge. | |
| Feb 27, 2014 at 0:10 | answer | added | Ian Agol | timeline score: 4 | |
| Feb 26, 2014 at 23:49 | comment | added | YCor | I checked twice: it's a deep theorem of Zelmanov (improving the solution of the restricted Burnside theorem) that every periodic profinite group is locally finite. (see E. Zelmanov, On periodic compact groups, Israel J. Math, 77 1992 83-95.) Thus the question indeed boils down to the post linked by Agol (although I wouldn't call it be called a duplicate). | |
| Feb 26, 2014 at 23:08 | comment | added | YCor | @Benjamin In Agol's link the assumption is weaker, namely that every element has finite order. Unless you can easily prove that a torsion profinite group is locally finite, the current question is a priori easier. | |
| Feb 26, 2014 at 22:02 | comment | added | Benjamin Steinberg | This question is essentially a duplicate of the one Agol has linked to. | |
| Feb 26, 2014 at 20:34 | comment | added | eric | An infinite product of cyclic groups of order 2 (with the product topology) is a vector space over the field with 2 elements, so it's locally finite and compact but certainly not finite. | |
| Feb 26, 2014 at 18:14 | comment | added | Ian Agol | This question is related: mathoverflow.net/a/459/1345 | |
| Feb 26, 2014 at 17:21 | comment | added | YCor | oh btw I misread since you assume compact from the beginning. Anyway the question remains: are there any examples other than closed subgroups of products of finite groups of bounded order. | |
| Feb 26, 2014 at 17:20 | comment | added | YCor | The topological analogue of LF for locally compact group is called (locally) elliptic; it means every (finite/compact) subset is contained in a compact open subgroup. If the group is abstractly locally finite, this is even stronger, but shows the study reduces to understanding which profinite groups are locally finite as abstract group. Right now I don't remember if there are other examples than the obvious ones (namely closed subgroups of products of finite groups of bounded size). | |
| Feb 26, 2014 at 17:03 | history | asked | Pablo | CC BY-SA 3.0 |