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This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): nilpotent group
View a complete list of such conjunctions

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Transitivity 3.2 Intermediate subgroup condition

Definition

A subgroup of a group is termed a nilpotent subnormal subgroup if it is nilpotent as a group and subnormal as a subgroup.

Relation with other properties

Stronger properties

• Abelian subnormal subgroup: Also related:
  • Abelian normal subgroup
  • Abelian characteristic subgroup
• Nilpotent normal subgroup: Also related:
  • Hereditarily normal subgroup
  • Nilpotent characteristic subgroup

Weaker properties

• Subnormal subgroup: Also related:
  • Hereditarily subnormal subgroup
  • Right-transitively fixed-depth subnormal subgroup
• Solvable subnormal subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: |
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties| View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: |
ABOUT INTERMEDIATE SUBROUP CONDITION: View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition