Characteristic subgroup of nilpotent group

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This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

The term characteristic subgroup of nilpotent group is used for a situation where the whole group is a nilpotent group and the subgroup is a characteristic subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of abelian group follows from abelian implies nilpotent follows from nilpotent not implies abelian |FULL LIST, MORE INFO
characteristic direct factor of nilpotent group Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
fully invariant subgroup of nilpotent group |FULL LIST, MORE INFO
verbal subgroup of nilpotent group Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
powering-invariant characteristic subgroup of nilpotent group characteristic not implies powering-invariant in nilpotent group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup of nilpotent group |FULL LIST, MORE INFO
nilpotent characteristic subgroup |FULL LIST, MORE INFO