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• Carter_subgroup abstract "In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups (Wehrfritz 1999).Carter (1961) proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the alternating group A5 of order 60 has no Carter subgroups. Vdovin (2006, 2007) showed that even if a finite group is not solvable then any two Carter subgroups are conjugate.A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups (Ballester-Bolinches & Ezquerro 2006, p. 100). For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgroups. Every subgroup containing a Carter subgroup of a soluble group is also self-normalizing, and a soluble group is generated by any Carter subgroup and its nilpotent residual (Schenkman 1975, VII.4.a).(Gaschütz 1963) viewed the Carter subgroups as analogues of Sylow subgroups and Hall subgroups, and unified their treatment with the theory of formations. In the language of formations, a Sylow p-subgroup is a covering group for the formation of p-groups, a Hall π-subgroup is a covering group for the formation of π-groups, and a Carter subgroup is a covering group for the formation of nilpotent groups (Ballester-Bolinches & Ezquerro 2006, p. 100). Together with an important generalization, Schunck classes, and an important dualization, Fischer classes, formations formed the major research themes of the late 20th century in the theory of finite soluble groups.A dual notion to Carter subgroups was introduced by Bernd Fischer in (Fischer 1966). A Fischer subgroup of a group is a nilpotent subgroup containing every other nilpotent subgroup it normalizes. A Fischer subgroup is a maximal nilpotent subgroup, but not every maximal nilpotent subgroup is a Fischer subgroup: again the nonabelian group of order six provides an example as every non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup (Wehrfritz 1999, p. 98).".
• Carter_subgroup wikiPageID "5456815".
• Carter_subgroup wikiPageLength "5289".
• Carter_subgroup wikiPageOutDegree "27".
• Carter_subgroup wikiPageRevisionID "665854484".
• Carter_subgroup wikiPageWikiLink Alternating_group.
• Carter_subgroup wikiPageWikiLink Bernd_Fischer_(mathematician).
• Carter_subgroup wikiPageWikiLink Cartan_subalgebra.
• Carter_subgroup wikiPageWikiLink Cartan_subgroup.
• Carter_subgroup wikiPageWikiLink Category:Finite_groups.
• Carter_subgroup wikiPageWikiLink Category:Solvable_groups.
• Carter_subgroup wikiPageWikiLink Category:Subgroup_properties.
• Carter_subgroup wikiPageWikiLink Central_series.
• Carter_subgroup wikiPageWikiLink Centralizer_and_normalizer.
• Carter_subgroup wikiPageWikiLink Dihedral_group_of_order_6.
• Carter_subgroup wikiPageWikiLink Finite_group.
• Carter_subgroup wikiPageWikiLink Formation_(group_theory).
• Carter_subgroup wikiPageWikiLink Group_theory.
• Carter_subgroup wikiPageWikiLink Habilitation.
• Carter_subgroup wikiPageWikiLink Habilitationsschrift.
• Carter_subgroup wikiPageWikiLink Hall_subgroup.
• Carter_subgroup wikiPageWikiLink Inner_automorphism.
• Carter_subgroup wikiPageWikiLink Mathematics.
• Carter_subgroup wikiPageWikiLink Mathematische_Zeitschrift.
• Carter_subgroup wikiPageWikiLink Nilpotent_group.
• Carter_subgroup wikiPageWikiLink Nilpotent_residual.
• Carter_subgroup wikiPageWikiLink Normalizer_condition.
• Carter_subgroup wikiPageWikiLink Roger_Carter_(mathematician).
• Carter_subgroup wikiPageWikiLink Self-normalizing_subgroup.
• Carter_subgroup wikiPageWikiLink Solvable_group.
• Carter_subgroup wikiPageWikiLink Springer-Verlag.
• Carter_subgroup wikiPageWikiLink Springer_Science+Business_Media.
• Carter_subgroup wikiPageWikiLink Sylow_subgroup.
• Carter_subgroup wikiPageWikiLink Sylow_theorems.
• Carter_subgroup wikiPageWikiLinkText "Carter subgroup".
• Carter_subgroup wikiPageWikiLinkText "formation".
• Carter_subgroup author "Vil'yams, N. N.".
• Carter_subgroup hasPhotoCollection Carter_subgroup.
• Carter_subgroup id "C/c020590".
• Carter_subgroup last "Vdovin".
• Carter_subgroup title "Carter subgroup".
• Carter_subgroup txt "yes".
• Carter_subgroup wikiPageUsesTemplate Template:Abstract-algebra-stub.
• Carter_subgroup wikiPageUsesTemplate Template:Citation.
• Carter_subgroup wikiPageUsesTemplate Template:Harv.
• Carter_subgroup wikiPageUsesTemplate Template:Harvs.
• Carter_subgroup wikiPageUsesTemplate Template:Harvtxt.
• Carter_subgroup wikiPageUsesTemplate Template:Springer.
• Carter_subgroup year "2006".
• Carter_subgroup year "2007".
• Carter_subgroup subject Category:Finite_groups.
• Carter_subgroup subject Category:Solvable_groups.
• Carter_subgroup subject Category:Subgroup_properties.
• Carter_subgroup hypernym Subgroup.
• Carter_subgroup type EthnicGroup.
• Carter_subgroup type Group.
• Carter_subgroup type Group.
• Carter_subgroup type Property.
• Carter_subgroup comment "In mathematics, especially in the field of group theory, a Carter subgroup of a finite group G is a self-normalizing subgroup of G that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups (Wehrfritz 1999).Carter (1961) proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic).".
• Carter_subgroup label "Carter subgroup".
• Carter_subgroup sameAs m.0dmqx4.
• Carter_subgroup sameAs Q5047226.
• Carter_subgroup sameAs Q5047226.
• Carter_subgroup wasDerivedFrom Carter_subgroup?oldid=665854484.
• Carter_subgroup isPrimaryTopicOf Carter_subgroup.