Talk:Pro-finite group

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Something perhaps needs to be said about the inverse systems? I mean, these are supposed to be indexed in such a way as to have a 'filtering' property? What is said about ind-finite isn't correct, anyway, unless the direct system tells you more than cyclic subgroups being finite, which is just a torsion group.

Charles Matthews 17:58, 12 Apr 2005 (UTC)

What it seems to amount to categorically is that a profinite group is a cofiltered limit of finite groups where the cofiltering category is also a partial order, and an ind-finite group is a filtered colimit where the filtering category is a partial order. The "inverse system" and "inductive system" terminology appears to be archaic from this point of view. In fact, some people even call the relevant functors "profinite groups", although this is not standard. - Gauge 05:39, 16 July 2006 (UTC)

[edit] natural topology

A profinite group, as defined here, doesn't have a natural topology until one selects an isomorphism with an inverse limit. It is still unclear whether the topology of a profinite group is unique, hence it's not proven that the given definitions are equivalent (i.e. that they define equivalent categories). This question is equivalent to the following:

Is every profinite group isomorphic to its profinite completion?

A group for which this holds is called strongly complete.

Zhw 00:52, 2 February 2006 (UTC)