Characteristic direct factor of abelian 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: 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