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Statements

Subject Item
dbr:Descendant_tree_(group_theory)
rdf:type
yago:Abstraction100002137 yago:Group100031264 yago:WikicatTopologicalGroups
rdfs:label
Descendant tree (group theory)
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.
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n7:Terminology_of_trees.png n7:TreeOf3Groups.png n7:TreeOf3Groups39.png n7:TreeOf2Groups.png n7:TreeOf2Groups222.png
dcterms:subject
dbc:P-groups dbc:Subgroup_series dbc:Group_theory dbc:Trees_(data_structures) dbc:Topological_groups
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1043708899
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n4:GraphOf3Groups.tiff dbr:Descendant_tree_(group_theory) dbc:Topological_groups dbr:Nilpotent_group dbr:Serial_module dbr:Magma_(computer_algebra_system) dbr:Tree_(set_theory) dbr:Commutator_subgroup dbr:Tree_(graph_theory) dbr:Space_group n4:Terminology_of_trees.png dbr:Giuseppe_Bagnera_(mathematician) dbr:Cohomology dbr:Presentation_of_a_group dbr:Tree_structure n4:TreeOf3Groups39.png dbr:Artin_transfer_(group_theory) dbr:Pro-p_group dbr:Quotient_group dbr:Mathematical_analysis dbr:Algebra n4:TreeOf2Groups.png dbr:P-group n4:TreeOf2Groups222.png dbc:Trees_(data_structures) n4:TreeOf3Groups.png dbr:Central_series dbc:Group_theory dbr:Group_theory dbr:Normal_subgroup dbc:Subgroup_series dbr:P-group_generation_algorithm dbr:Topological_group dbr:Isomorphism_class dbr:GAP_(computer_algebra_system) dbc:P-groups
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n7:Terminology_of_trees.png?width=300
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Proof
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background:LightSteelBlue; text-align:center;
dbo: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.
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wikipedia-en:Descendant_tree_(group_theory)