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In mathematics, a pro-p group (for some prime number p) is a profinite group such that for any open normal subgroup the quotient group is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of discrete finite p-groups.

Attributes	Values
rdf:type	yago:WikicatTopologicalGroups yago:Abstraction100002137 yago:Group100031264
rdfs:label	Pro-p group (en)
rdfs:comment	In mathematics, a pro-p group (for some prime number p) is a profinite group such that for any open normal subgroup the quotient group is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of discrete finite p-groups. (en)
dct:subject	Properties of groups Topological groups Infinite group theory P-groups
Wikipage page ID	1842075 (xsd:integer)
Wikipage revision ID	979061193 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Descendant tree (group theory) Index of a subgroup Profinite group Compact space Mathematics Matrix (mathematics) Normal subgroup Quotient group Closed set Identity matrix Michel Lazard Properties of groups Topological groups Alexander Lubotzky Infinite group theory P-adic number Residual property (mathematics) Hilbert's fifth problem Inverse limit Prime number P-groups Discrete space Manifold Open set Inverse system P-group Prüfer rank
sameAs	Pro-p group Pro-p group Pro-p group Pro-p group
dbp:wikiPageUsesTemplate	dbt:Algebra-stub dbt:Citation
has abstract	In mathematics, a pro-p group (for some prime number p) is a profinite group such that for any open normal subgroup the quotient group is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of discrete finite p-groups. The best-understood (and historically most important) class of pro-p groups is the p-adic analytic groups: groups with the structure of an analytic manifold over such that group multiplication and inversion are both analytic functions.The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i.e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements. More generally it was shown that a finitely generated profinite group is a compact p-adic Lie group if and only if it has an open subgroup that is a uniformly powerful pro-p-group. The Coclass Theorems have been proved in 1994 by A. Shalev and independently by C. R. Leedham-Green. Theorem D is one of these theorems and asserts that, for any prime number p and any positive integer r, there exist only finitely many pro-p groups of coclass r. This finiteness result is fundamental for the classification of finite p-groups by means of directed coclass graphs. (en)
prov:wasDerivedFrom	wikipedia-en:Pro-p_group?oldid=979061193&ns=0
page length (characters) of wiki page	3786 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Pro-p_group
is Link from a Wikipage to another Wikipage of	Charles Leedham-Green Descendant tree (group theory) Iwasawa algebra Profinite group Nottingham group Georgia Benkart Demushkin group Ju-Lee Kim Residual property (mathematics) Principal ideal theorem Howson property Powerful p-group P-adic analytic group P-group Prüfer rank Pro-p-group
is Wikipage redirect of	P-adic analytic group Pro-p-group
is foaf:primaryTopic of	wikipedia-en:Pro-p_group

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