Characteristic direct factor of nilpotent group

This article describes a property that arises as the conjunction of a subgroup property: characteristic direct factor 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 $H$ of a group $G$ is termed a characteristic direct factor of nilpotent group if it satisfies the following equivalent conditions:

1. $G$ is a nilpotent group and $H$ is a characteristic direct factor of $G$ (i.e., $H$ is both a characteristic subgroup of $G$ and a direct factor of $G$).
2. $G$ is a nilpotent group and $H$ is a fully invariant direct factor of $G$ (i.e., $H$ is both a fully invariant subgroup of $G$ and a direct factor of $G$). This has other equivalent formulations; see equivalence of definitions of fully invariant direct factor.

Equivalence of definitions

Further information: equivalence of definitions of characteristic direct factor of nilpotent group

The equivalence follows indirectly from the fact that nontrivial subgroup of nilpotent group has nontrivial homomorphism to center.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic direct factor of abelian group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant direct factor |FULL LIST, MORE INFO
characteristic direct factor Fully invariant direct factor|FULL LIST, MORE INFO
fully invariant subgroup of nilpotent group |FULL LIST, MORE INFO
characteristic subgroup of nilpotent group Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
direct factor of nilpotent group |FULL LIST, MORE INFO
fully invariant subgroup Fully invariant direct factor, Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO
characteristic subgroup Fully invariant direct factor|FULL LIST, MORE INFO
direct factor Fully invariant direct factor|FULL LIST, MORE INFO