About: Descendant tree (group theory)

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In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order , for a fixed prime number and varying integer exponents . Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups.

Attributes	Values
rdf:type	yago:WikicatTopologicalGroups yago:Abstraction100002137 yago:Group100031264
rdfs:label	Descendant tree (group theory) (en)
rdfs:comment	In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order , for a fixed prime number and varying integer exponents . Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups. (en)
foaf:depiction
dcterms:subject	Group theory Topological groups Subgroup series Trees (data structures) P-groups
Wikipage page ID	42823620 (xsd:integer)
Wikipage revision ID	1043708899 (xsd:integer)
Link from a Wikipage to another Wikipage	Presentation of a group Pro-p group Descendant tree (group theory) Group theory Mathematical analysis Normal subgroup Serial module Quotient group GAP (computer algebra system) Giuseppe Bagnera (mathematician) Magma (computer algebra system) Commutator subgroup Topological groups Topological group Tree (graph theory) Tree (set theory) Algebra Nilpotent group Subgroup series Central series Isomorphism class Tree structure Artin transfer (group theory) Trees (data structures) P-groups Cohomology Space group Group theory P-group P-group generation algorithm
sameAs	Descendant tree (group theory) Descendant tree (group theory) Descendant tree (group theory) Descendant tree (group theory)
dbp:wikiPageUsesTemplate	dbt:Hidden_begin dbt:Hidden_end dbt:Reflist dbt:EquationNote
thumbnail	wiki-commons:Special:FilePath/Terminology_of_trees.png?width=300
title	Proof (en)
titlestyle	background:LightSteelBlue; text-align:center; (en)
has abstract	In mathematics, specifically group theory, a descendant tree is a hierarchical structure that visualizes parent-descendant relations between isomorphism classes of finite groups of prime power order , for a fixed prime number and varying integer exponents . Such groups are briefly called finite p-groups. The vertices of a descendant tree are isomorphism classes of finite p-groups. Additionally to their order , finite p-groups have two further related invariants, the nilpotency class and the coclass . It turned out that descendant trees of a particular kind, the so-called pruned coclass trees whose infinitely many vertices share a common coclass , reveal a repeating finite pattern. These two crucial properties of finiteness and periodicity admit a characterization of all members of the tree by finitely many parametrized presentations. Consequently, descendant trees play a fundamental role in the classification of finite p-groups. By means of kernels and targets of Artin transfer homomorphisms, descendant trees can be endowed with additional structure. An important question is how the descendant tree can actually be constructed for an assigned starting group which is taken as the root of the tree. The p-group generation algorithm is a recursive process for constructing the descendant tree of a given finite p-group playing the role of the tree root. This algorithm is implemented in the computational algebra systems GAP and Magma. (en)
prov:wasDerivedFrom	wikipedia-en:Descendant_tree_(group_theory)?oldid=1043708899&ns=0
page length (characters) of wiki page	48812 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Descendant_tree_(group_theory)
is Link from a Wikipage to another Wikipage of	Pro-p group Descendant tree (group theory) Giuseppe Bagnera (mathematician) Central series Tree structure Artin transfer (group theory) Coclass Principal ideal theorem Mihály Nagy P-group P-group generation algorithm
is foaf:primaryTopic of	wikipedia-en:Descendant_tree_(group_theory)

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