Talk:Prüfer rank

Definition over-specialised?
The definition seems overly specialised to the case of pro-p-groups. The literature tends to favour the definition that a group has Prüfer rank r if every finitely generated subgroup requires at most r generators, and r is minimal. See for example, Are these definitions consistent? Is there a reliable source relating them? Deltahedron (talk) 19:56, 21 June 2014 (UTC)
 * Additional. There is a further definition of Prüfer rank for pro-p-groups, namely the smallest d such that every subgroup of a finite toplogical quotient is d-generated, at .  Deltahedron (talk) 11:29, 22 June 2014 (UTC)
 * I don't think I have the expertise to give a useful opinion on what the best definition and level of generality of Prüfer rank should be. But here's the definition from the Yamagishi ref I added: he defines the Prüfer rank of a pro-p-group G using the formula
 * $$\operatorname{rk}(G)=\overline\lim\{d(U)\mid U<_oG\}.$$
 * Here $$<_o$$ means "is an open subgroup of" and
 * $$d(G)=\dim_{\mathbb{Z}/p\mathbb{Z}}H^1(G,\mathbb{Z}/p\mathbb{Z})$$
 * ("the minimal number of topological generators", assumed to be finite). I guess this is actually closer to Nikolov's definition rather than as I thought using abelian rank of a quotient group? I don't think I checked carefully enough whether Yamagishi was using the same d as the one here. But I think it comes out to the same number. —David Eppstein (talk) 18:12, 22 June 2014 (UTC)
 * I agree that the Yamagishi definition is pretty close to that of Nikolov. But neither seems obviously related to that given in the article.
 * A reference for the assertion "finite Prüfer rank is equivalent to having an open normal subgroup that is powerful. In turn these are precisely the class of pro-p groups that are p-adic analytic" seems to be  The definition of r in that paper seems to be that of Yamamgishi.  Deltahedron (talk) 18:36, 22 June 2014 (UTC)