# Characteristic direct factor of nilpotent group

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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