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,

• Lennox, John C.; Robinson, Derek J. S. (2004). The Theory of Infinite Soluble Groups. Oxford Mathematical Monographs. Oxford University Press. p. 85. ISBN 0-19-152315-1.
• Dixon, Martyn Russell (1994). Sylow Theory, Formations, and Fitting Classes in Locally Finite Groups. Series in algebra. World Scientific. p. 44. ISBN 9810217951.
• Campbell, C. M.; Quick, M. R.; Robertson, E. F.; Roney-Dougal, C. M.; Smith, G. C.; Traustason, G., eds. (2011). Groups St Andrews 2009 in Bath. London Mathematical Society Lecture Note Series. Vol. 387. Cambridge University Press. p. 104. ISBN 1139498274.

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 Nikolov, Nikolay (22 Feb 2012). "Algebraic properties of profinite groups". arXiv:1108.5130 [math.GR]. {{cite arXiv}}: Unknown parameter |version= ignored (help). 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

$\operatorname {rk} (G)={\overline {\lim\{}}d(U)\mid U<_{o}G\}.$

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 Lubotzky, Alexander; Mann, Avinoam (1987). "Powerful p-groups. II: p-adic analytic groups". J. Algebra. 105: 506–515. doi:10.1016/0021-8693(87)90212-2. Zbl 0626.20022. The definition of r in that paper seems to be that of Yamamgishi. Deltahedron (talk) 18:36, 22 June 2014 (UTC)