Fully invariant subgroup of nilpotent group

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This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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

A subgroup of a group is termed a fully invariant subgroup of nilpotent group if the whole group is a nilpotent group and the subgroup is a fully invariant subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup of abelian group abelian implies nilpotent follows from nilpotent not implies abelian (use the group as a subgroup of itself) |FULL LIST, MORE INFO
verbal subgroup of nilpotent group follows from verbal implies fully invariant follows from fully invariant not implies verbal in finite abelian group |FULL LIST, MORE INFO
characteristic direct factor of nilpotent group |FULL LIST, MORE INFO
verbal subgroup of abelian group follows by combining abelian implies nilpotent and verbal implies fully invariant follows from nilpotent not implies abelian (use the group as a subgorup of itself) Verbal subgroup of nilpotent group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of nilpotent group the subgroup is a characteristic subgroup and the whole group is nilpotent follows from fully invariant implies characteristic follows from characteristic not implies fully invariant in finite abelian group |FULL LIST, MORE INFO
normal subgroup of nilpotent group the subgroup is a normal subgroup and the whole group is nilpotent (via characteristic) (via characteristic) Characteristic subgroup of nilpotent group|FULL LIST, MORE INFO
subgroup of nilpotent group the whole group is nilpotent |FULL LIST, MORE INFO
fully invariant subgroup invariant under all endomorphisms any non-nilpotent group as a subgroup of itself |FULL LIST, MORE INFO