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Statements

Subject Item
dbr:Central_series
rdfs:label
Central series
rdfs:comment
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras.
dcterms:subject
dbc:Subgroup_series
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15202655
dbo:wikiPageRevisionID
1070866444
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dbr:Group_ring dbr:Loewy_series dbr:Centerless_group dbr:Mathematics dbr:Residually_nilpotent dbr:Commutator_subgroup dbr:Generating_set_of_a_group dbr:Group_theory dbr:Unipotent_group dbr:Direct_product dbr:Engel's_theorem dbr:Frattini_subgroup dbr:Adjoint_representation_of_a_Lie_group dbr:Lie_theory dbr:Group_(mathematics) dbr:Lie_subalgebra dbr:Lower_Fitting_series dbr:Combinatorial_group_theory dbr:Grün's_lemma dbr:Commutator dbr:Cyclic_group dbr:Free_group dbr:Direct_sum_of_groups dbr:Ordinal_numbers dbr:Symmetric_group dbr:Solvable_group dbr:Center_(group_theory) dbr:Elementary_abelian_group dbr:Flag_(linear_algebra) dbr:Quaternion_group dbr:Socle_(mathematics) dbr:Sylow_subgroup dbr:P-group dbr:Limit_ordinal dbr:Perfect_group dbr:Periodic_group dbr:Exponent_(group_theory) dbr:Nilpotent_group dbr:Matrix_ring dbr:Upper_triangular dbr:Subgroup dbr:Derived_series dbr:Stephen_Arthur_Jennings dbr:Factor_group dbr:Normal_series dbr:Transfinite_recursion dbc:Subgroup_series dbr:Descendant_tree_(group_theory) dbr:Derived_subgroup dbr:Nilpotent_series dbr:Center_(group)
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dbo:abstract
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras. The lower central series and upper central series (also called the descending central series and ascending central series, respectively), are, despite the "central" in their names, central series if and only if a group is nilpotent.
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