# Characteristic direct factor of abelian 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: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
This article describes a property that arises as the conjunction of a subgroup property: fully invariant direct factor with a group property imposed on the ambient group: abelian 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 $G$ if the following equivalent conditions are satisfied:

1. $G$ is an abelian group and $H$ is a characteristic direct factor of $G$ (i.e., $H$ is both a characteristic subgroup of $G$ and a direct factor in $G$).
2. $G$ is an abelian group and $H$ is a fully invariant direct factor of $G$ (i.e., $H$ is a fully invariant subgroup as well as a direct factor of $G$). See also equivalence of definitions of fully invariant direct factor for other equivalent formulations of this.

### Equivalence of definitions

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

## Relation with other properties

### Weaker properties

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